String’s Depth of Burial

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A string’s depth might be defined as the execution time of its minimal program.

The difficulty with this definition arises in cases where the minimal program is only a few bits smaller than some much faster program, such as a print program, to compute the same output x. In this case, slight changes in x may induce arbitrarily large changes in the run time of the minimal program, by changing which of the two competing programs is minimal. Analogous instability manifests itself in translating programs from one universal machine to another. This instability emphasizes the essential role of the quantity of buried redundancy, not as a measure of depth, but as a certifier of depth. In terms of the philosophy-of-science metaphor, an object whose minimal program is only a few bits smaller than its print program is like an observation that points to a nontrivial hypothesis, but with only a low level of statistical confidence.

To adequately characterize a finite string’s depth one must therefore consider the amount of buried redundancy as well as the depth of its burial. A string’s depth at significance level s might thus be defined as that amount of time complexity which is attested by s bits worth of buried redundancy. This characterization of depth may be formalized in several ways.

A string’s depth at significance level s be defined as the time required to compute the string by a program no more than s bits larger than the minimal program.

This definition solves the stability problem, but is unsatisfactory in the way it treats multiple programs of the same length. Intuitively, 2k distinct (n + k)-bit programs that compute same output ought to be accorded the same weight as one n-bit program; but, by the present definition, they would be given no more weight than one (n + k)-bit program.

A string’s depth at signicifcance level s depth might be defined as the time t required for the string’s time-bounded algorithmic probability Pt(x) to rise to within a factor 2−s of its asymptotic time-unbounded value P(x).

This formalizes the notion that for the string to have originated by an effective process of t steps or fewer is less plausible than for the first s tosses of a fair coin all to come up heads.

It is not known whether there exist strings that are deep, in other words, strings having no small fast programs, even though they have enough large fast programs to contribute a significant fraction of their algorithmic probability. Such strings might be called deterministically deep but probabilistically shallow, because their chance of being produced quickly in a probabilistic computation (e.g. one where the input bits of U are supplied by coin tossing) is significant compared to their chance of being produced slowly. The question of whether such strings exist is probably hard to answer because it does not relativize uniformly. Deterministic and probabilistic depths are not very different relative to a random coin-toss oracle A of the equality of random-oracle-relativized deterministic and probabilistic polynomial time complexity classes; but they can be very different relative to an oracle B deliberately designed to hide information from deterministic computations (this parallels Hunt’s proof that deterministic and probabilistic polynomial time are unequal relative to such an oracle).

(Depth of Finite Strings): Let x and w be strings and s a significance parameter. A string’s depth at significance level s, denoted Ds(x), will be defined as min{T(p) : (|p|−|p| < s)∧(U(p) = x)}, the least time required to compute it by a s-incompressible program. At any given significance level, a string will be called t-deep if its depth exceeds t, and t-shallow otherwise.

The difference between this definition and the previous one is rather subtle philosophically and not very great quantitatively. Philosophically, when each individual hypothesis for the rapid origin of x is implausible at the 2−s confidence level, then it requires only that a weighted average of all such hypotheses be implausible.

There exist constants c1 and c2 such that for any string x, if programs running in time ≤ t contribute a fraction between 2−s and 2−s+1 of the string’s total algorithmic probability, then x has depth at most t at significance level s + c1 and depth at least t at significance level s − min{H(s), H(t)} − c2.

Proof : The first part follows easily from the fact that any k-compressible self-delimiting program p is associated with a unique, k − O(1) bits shorter, program of the form “execute the result of executing p∗”. Therefore there exists a constant c1 such that if all t-fast programs for x were s + c1– compressible, the associated shorter programs would contribute more than the total algorithmic probability of x. The second part follows because, roughly, if fast programs contribute only a small fraction of the algorithmic probability of x, then the property of being a fast program for x is so unusual that no program having that property can be random. More precisely, the t-fast programs for x constitute a finite prefix set, a superset S of which can be computed by a program of size H(x) + min{H(t), H(s)} + O(1) bits. (Given x∗ and either t∗ or s∗, begin enumerating all self-delimiting programs that compute x, in order of increasing running time, and quit when either the running time exceeds t or the accumulated measure of programs so far enumerated exceeds 2−(H(x)−s)). Therefore there exists a constant c2 such that, every member of S, and thus every t-fast program for x, is compressible by at least s − min{H(s), H(t)} − O(1) bits.

The ability of universal machines to simulate one another efficiently implies a corresponding degree of machine-independence for depth: for any two efficiently universal machines of the sort considered here, there exists a constant c and a linear polynomial L such that for any t, strings whose (s+c)-significant depth is at least L(t) on one machine will have s-significant depth at least t on the other.

Depth of one string relative to another may be defined analogously, and represents the plausible time required to produce one string, x, from another, w.

(Relative Depth of Finite Strings): For any two strings w and x, the depth of x relative to w at significance level s, denoted Ds(x/w), will be defined as min{T(p, w) : (|p|−|(p/w)∗| < s)∧(U(p, w) = x)}, the least time required to compute x from w by a program that is s-incompressible relative to w.

Depth of a string relative to its length is a particularly useful notion, allowing us, as it were, to consider the triviality or nontriviality of the “content” of a string (i.e. its bit sequence), independent of its “form” (length). For example, although the infinite sequence 000… is intuitively trivial, its initial segment 0n is deep whenever n is deep. However, 0n is always shallow relative to n, as is, with high probability, a random string of length n.

In order to adequately represent the intuitive notion of stored mathematical work, it is necessary that depth obey a “slow growth” law, i.e. that fast deterministic processes be unable to transform a shallow object into a deep one, and that fast probabilistic processes be able to do so only with low probability.

(Slow Growth Law): Given any data string x and two significance parameters s2 > s1, a random program generated by coin tossing has probability less than 2−(s2−s1)+O(1) of transforming x into an excessively deep output, i.e. one whose s2-significant depth exceeds the s1-significant depth of x plus the run time of the transforming program plus O(1). More precisely, there exist positive constants c1, c2 such that for all strings x, and all pairs of significance parameters s2 > s1, the prefix set {q : Ds2(U(q, x)) > Ds1(x) + T(q, x) + c1} has measure less than 2−(s2−s1)+c2.

Proof: Let p be a s1-incompressible program which computes x in time Ds1(x), and let r be the restart prefix mentioned in the definition of the U machine. Let Q be the prefix set {q : Ds2(U(q, x)) > T(q, x) + Ds1(x) + c1}, where the constant c1 is sufficient to cover the time overhead of concatenation. For all q ∈ Q, the program rpq by definition computes some deep result U(q, x) in less time than that result’s own s2-significant depth, and so rpq must be compressible by s2 bits. The sum of the algorithmic probabilities of strings of the form rpq, where q ∈ Q, is therefore

Σq∈Q P(rpq)< Σq∈Q 2−|rpq| + s2 = 2−|r|−|p|+s2 μ(Q)

On the other hand, since the self-delimiting program p can be recovered from any string of the form rpq (by deleting r and executing the remainder pq until halting occurs, by which time exactly p will have been read), the algorithmic probability of p is at least as great (within a constant factor) as the sum of the algorithmic probabilities of the strings {rpq : q ∈ Q} considered above:

P(p) > μ(Q) · 2−|r|−|p|+s2−O(1)

Recalling the fact that minimal program size is equal within a constant factor to the −log of algorithmic probability, and the s1-incompressibility of p, we have P(p) < 2−(|p|−s1+O(1)), and therefore finally

μ(Q) < 2−(s2−s1)+O(1), which was to be demonstrated.

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Pareto Optimality

There are some solutions. (“If you don’t give a solution, you are part of the problem”). Most important: Human wealth should be set as the only goal in society and economy. Liberalism is ruinous for humans, while it may be optimal for fitter entities. Nobody is out there to take away the money of others without working for it. In a way of ‘revenge’ or ‘envy’, (basically justifying laziness) taking away the hard-work earnings of others. No way. Nobody wants it. Thinking that yours can be the only way a rational person can think. Anybody not ‘winning’ the game is a ‘loser’. Some of us, actually, do not even want to enter the game.

Yet – the big dilemma – that money-grabbing mentality is essential for the economy. Without it we would be equally doomed. But, what we will see now is that you’ll will lose every last penny either way, even without divine intervention.

Having said that, the solution is to take away the money. Seeing that the system is not stable and accumulates the capital on a big pile, disconnected from humans, mathematically there are two solutions:

1) Put all the capital in the hands of people. If profit is made M’-M, this profit falls to the hands of the people that caused it. This seems fair, and mathematically stable. However, how the wealth is then distributed? That would be the task of politicians, and history has shown that they are a worse pest than capital. Politicians, actually, always wind up representing the capital. No country in the world ever managed to avoid it.

2) Let the system be as it is, which is great for giving people incentives to work and develop things, but at the end of the year, redistribute the wealth to follow an ideal curve that optimizes both wealth and increments of wealth.

The latter is an interesting idea. Also since it does not need rigorous restructuring of society, something that would only be possible after a total collapse of civilization. While unavoidable in the system we have, it would be better to act pro-actively and do something before it happens. Moreover, since money is air – or worse, vacuum – there is actually nothing that is ‘taken away’. Money is just a right to consume and can thus be redistributed at will if there is a just cause to do so. In normal cases this euphemistic word ‘redistribution’ amounts to theft and undermines incentives for work and production and thus causes poverty. Yet, if it can be shown to actually increase incentives to work, and thus increase overall wealth, it would need no further justification.

We set out to calculate this idea. However, it turned out to give quite remarkable results. Basically, the optimal distribution is slavery. Let us present them here. Let’s look at the distribution of wealth. Figure below shows a curve of wealth per person, with the richest conventionally placed at the right and the poor on the left, to result in what is in mathematics called a monotonously-increasing function. This virtual country has 10 million inhabitants and a certain wealth that ranges from nearly nothing to millions, but it can easily be mapped to any country.

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Figure 1: Absolute wealth distribution function

As the overall wealth increases, it condenses over time at the right side of the curve. Left unchecked, the curve would become ever-more skew, ending eventually in a straight horizontal line at zero up to the last uttermost right point, where it shoots up to an astronomical value. The integral of the curve (total wealth/capital M) always increases, but it eventually goes to one person. Here it is intrinsically assumed that wealth, actually, is still connected to people and not, as it in fact is, becomes independent of people, becomes ‘capital’ autonomously by itself. If independent of people, this wealth can anyway be without any form of remorse whatsoever be confiscated and redistributed. Ergo, only the system where all the wealth is owned by people is needed to be studied.

A more interesting figure is the fractional distribution of wealth, with the normalized wealth w(x) plotted as a function of normalized population x (that thus runs from 0 to 1). Once again with the richest plotted on the right. See Figure below.

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Figure 2: Relative wealth distribution functions: ‘ideal communist’ (dotted line. constant distribution), ‘ideal capitalist’ (one person owns all, dashed line) and ‘ideal’ functions (work-incentive optimized, solid line).

Every person x in this figure feels an incentive to work harder, because it wants to overtake his/her right-side neighbor and move to the right on the curve. We can define an incentive i(x) for work for person x as the derivative of the curve, divided by the curve itself (a person will work harder proportional to the relative increase in wealth)

i(x) = dw(x)/dx/w(x) —– (1)

A ‘communistic’ (in the negative connotation) distribution is that everybody earns equally, that means that w(x) is constant, with the constant being one

‘ideal’ communist: w(x) = 1.

and nobody has an incentive to work, i(x) = 0 ∀ x. However, in a utopic capitalist world, as shown, the distribution is ‘all on a big pile’. This is what mathematicians call a delta-function

‘ideal’ capitalist: w(x) = δ(x − 1),

and once again, the incentive is zero for all people, i(x) = 0. If you work, or don’t work, you get nothing. Except one person who, working or not, gets everything.

Thus, there is somewhere an ‘ideal curve’ w(x) that optimizes the sum of incentives I defined as the integral of i(x) over x.

I = ∫01i(x)dx = ∫01(dw(x)/dx)/w(x) dx = ∫x=0x=1dw(x)/w(x) = ln[w(x)]|x=0x=1 —– (2)

Which function w is that? Boundary conditions are

1. The total wealth is normalized: The integral of w(x) over x from 0 to 1 is unity.

01w(x)dx = 1 —– (3)

2. Everybody has a at least a minimal income, defined as the survival minimum. (A concept that actually many societies implement). We can call this w0, defined as a percentage of the total wealth, to make the calculation easy (every year this parameter can be reevaluated, for instance when the total wealth increased, but not the minimum wealth needed to survive). Thus, w(0) = w0.

The curve also has an intrinsic parameter wmax. This represents the scale of the figure, and is the result of the other boundary conditions and therefore not really a parameter as such. The function basically has two parameters, minimal subsistence level w0 and skewness b.

As an example, we can try an exponentially-rising function with offset that starts by being forced to pass through the points (0, w0) and (1, wmax):

w(x) = w0 + (wmax − w0)(ebx −1)/(eb − 1) —– (4)

An example of such a function is given in the above Figure. To analytically determine which function is ideal is very complicated, but it can easily be simulated in a genetic algorithm way. In this, we start with a given distribution and make random mutations to it. If the total incentive for work goes up, we keep that new distribution. If not, we go back to the previous distribution.

The results are shown in the figure 3 below for a 30-person population, with w0 = 10% of average (w0 = 1/300 = 0.33%).

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Figure 3: Genetic algorithm results for the distribution of wealth (w) and incentive to work (i) in a liberal system where everybody only has money (wealth) as incentive. 

Depending on the starting distribution, the system winds up in different optima. If we start with a communistic distribution of figure 2, we wind up with a situation in which the distribution stays homogeneous ‘everybody equal’, with the exception of two people. A ‘slave’ earns the minimum wages and does nearly all the work, and a ‘party official’ that does not do much, but gets a large part of the wealth. Everybody else is equally poor (total incentive/production equal to 21), w = 1/30 = 10w0, with most people doing nothing, nor being encouraged to do anything. The other situation we find when we start with a random distribution or linear increasing distribution. The final situation is shown in situation 2 of the figure 3. It is equal to everybody getting minimum wealth, w0, except the ‘banker’ who gets 90% (270 times more than minimum), while nobody is doing anything, except, curiously, the penultimate person, which we can call the ‘wheedler’, for cajoling the banker into giving him money. The total wealth is higher (156), but the average person gets less, w0.

Note that this isn’t necessarily an evolution of the distribution of wealth over time. Instead, it is a final, stable, distribution calculated with an evolutionary (‘genetic’) algorithm. Moreover, this analysis can be made within a country, analyzing the distribution of wealth between people of the same country, as well as between countries.

We thus find that a liberal system, moreover one in which people are motivated by the relative wealth increase they might attain, winds up with most of the wealth accumulated by one person who not necessarily does any work. This is then consistent with the tendency of liberal capitalist societies to have indeed the capital and wealth accumulate in a single point, and consistent with Marx’s theories that predict it as well. A singularity of distribution of wealth is what you get in a liberal capitalist society where personal wealth is the only driving force of people. Which is ironic, in a way, because by going only for personal wealth, nobody gets any of it, except the big leader. It is a form of Prisoner’s Dilemma.

Gothic: Once Again Atheistic Materialism and Hedonistic Flirtations. Drunken Risibility.

 

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The machinery of the Gothic, traditionally relegated to both a formulaic and a sensational aesthetic, gradually evolved into a recyclable set of images, motifs and narrative devices that surpass temporal, spatial and generic categories. From the moment of its appearance the Gothic has been obsessed with presenting itself as an imitation.

Recent literary theory has extensively probed into the power of the Gothic to evade temporal and generic limits and into the aesthetic, narratological and ideological implications this involves. Officially granting the Gothic the elasticity it has always entailed has resulted in a reconfiguration of its spectrum both synchronically – by acknowledging its influence on numerous postmodern fictions – and diachronically – by rescripting, in hindsight, the history of its canon so as to allow space for ambiguous presences.

Both transgressive and hybrid in form and content, the Gothic has been accepted as a malleable genre, flexible enough to create more freely, in Borgesian fashion, its own precursors. The genre flouted what are considered the basic principles of good prose writing: adherence to verisimilitude and avoidance of both narrative diversions and moralising – all of which are, of course, made to be deliberately upset. Many merely cite the epigrammatic power of the essay’s most renowned phrase, that the rise of the Gothic “was the inevitable result of the revolutionary shocks which all of Europe has suffered”.

The eighteenth-century French materialist philosophy purported the displacement of metaphysical investigations into the meaning of life by materialist explorations. Julien Offray de La Mettrie, a French physician and philosopher, the earliest of materialist writers of the Enlightenment, published the materialist manifesto L’ Homme machine (Man a Machine), that did away with the transcendentalism of the soul, banished all supernatural agencies by claiming that mind is as mechanical as matter and equated humans with machines. In his words: “The human body is a machine that winds up its own springs: it is a living image of the perpetual motion”. French materialist thought resulted in the publication of the great 28-volume Encyclopédie, ou Dictionnaire raisonné des sciences, des arts et des méttrie par une société de gens de lettres by Denis Diderot and Jean Le Rond d’ Alembert, and which was grounded on purely materialist principles, against all kinds of metaphysical thinking. Diderot’s atheist materialism set the tone of the Encyclopédie, which, for both editors, was the ideal vehicle […] for reshaping French high culture and attitudes, as well as the perfect instrument with which to insinuate their radical Weltanschauung surreptitiously, using devious procedures, into the main arteries of French Society, embedding their revolutionary philosophic manifesto in a vast compilation ostensibly designed to provide plain information and basic orientation but in fact subtly challenging and transforming attitudes in every respect. While materialist thinkers ultimately disowned La Mettrie because he ran counter to their systematic moral, political and social naturalism, someone like Sade remained deeply influenced and inspired for his indebtedness to La Mettrie’s atheism and hedonism, particularly to the perception of virtue and vice as relative notions − the result of socialisation and at odds with nature.

 

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

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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!

Cosmology: Friedmann-Lemaître Universes

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Cosmology starts by assuming that the large-scale evolution of spacetime can be determined by applying Einstein’s field equations of Gravitation everywhere: global evolution will follow from local physics. The standard models of cosmology are based on the assumption that once one has averaged over a large enough physical scale, isotropy is observed by all fundamental observers (the preferred family of observers associated with the average motion of matter in the universe). When this isotropy is exact, the universe is spatially homogeneous as well as isotropic. The matter motion is then along irrotational and shearfree geodesic curves with tangent vector ua, implying the existence of a canonical time-variable t obeying ua = −t,a. The Robertson-Walker (‘RW’) geometries used to describe the large-scale structure of the universe embody these symmetries exactly. Consequently they are conformally flat, that is, the Weyl tensor is zero:

Cijkl := Rijkl + 1/2(Rikgjl + Rjlgik − Ril gjk − Rjkgil) − 1/6R(gikgjl − gilgjk) = 0 —– (1)

this tensor represents the free gravitational field, enabling non-local effects such as tidal forces and gravitational waves which do not occur in the exact RW geometries.

Comoving coordinates can be chosen so that the metric takes the form:

ds2 = −dt2 + S2(t)dσ2, ua = δa0 (a=0,1,2,3) —– (2)

where S(t) is the time-dependent scale factor, and the worldlines with tangent vector ua = dxa/dt represent the histories of fundamental observers. The space sections {t = const} are surfaces of homogeneity and have maximal symmetry: they are 3-spaces of constant curvature K = k/S2(t) where k is the sign of K. The normalized metric dσ2 characterizes a 3-space of normalized constant curvature k; coordinates (r, θ, φ) can be chosen such that

2 = dr2 + f2(r) dθ2 + sin2θdφ2 —– (3)

where f (r) = {sin r, r, sinh r} if k = {+1, 0, −1} respectively. The rate of expansion at any time t is characterized by the Hubble parameter H(t) = S ̇/S.

To determine the metric’s evolution in time, one applies the Einstein Field Equations, showing the effect of matter on space-time curvature, to the metric (2,3). Because of local isotropy, the matter tensor Tab necessarily takes a perfect fluid form relative to the preferred worldlines with tangent vector ua:

Tab = (μ + p/c2)uaub + (p/c2)gab —– (4)

, c is the speed of light. The energy density μ(t) and pressure term p(t)/c2 are the timelike and spacelike eigenvalues of Tab. The integrability conditions for the Einstein Field Equations are the energy-density conservation equation

Tab;b = 0 ⇔ μ ̇ + (μ + p/c2)3S ̇/S = 0 —– (5)

This becomes determinate when a suitable equation of state function w := pc2/μ relates the pressure p to the energy density μ and temperature T : p = w(μ,T)μ/c2 (w may or may not be constant). Baryons have {pbar = 0 ⇔ w = 0} and radiation has {pradc2 = μrad/3 ⇔ w = 1/3,μrad = aT4rad}, which by (5) imply

μbar ∝ S−3, μrad ∝ S−4, Trad ∝ S−1 —– (6)

The scale factor S(t) obeys the Raychaudhuri equation

3S ̈/S = -1/2 κ(μ + 3p/c2) + Λ —– (7)

, where κ is the gravitational constant and Λ is the cosmological constant. A cosmological constant can also be regarded as a fluid with pressure p related to the energy density μ by {p = −μc2 ⇔ w = −1}. This shows that the active gravitational mass density of the matter and fields present is μgrav := μ + 3p/c2. For ordinary matter this will be positive:

μ + 3p/c2 > 0 ⇔ w > −1/3 —– (8)

(the ‘Strong Energy Condition’), so ordinary matter will tend to cause the universe to decelerate (S ̈ < 0). It is also apparent that a positive cosmological constant on its own will cause an accelerating expansion (S ̈ > 0). When matter and a cosmological constant are both present, either result may occur depending on which effect is dominant. The first integral of equations (5, 7) when S ̇≠ 0 is the Friedmann equation

S ̇2/S2 = κμ/3 + Λ/3 – k/S2 —– (9)

This is just the Gauss equation relating the 3-space curvature to the 4-space curvature, showing how matter directly causes a curvature of 3-spaces. Because of the spacetime symmetries, the ten Einstein Filed Equations are equivalent to the two equations (7, 9). Models of this kind, that is with a Robertson-Walker (‘RW’) geometry with metric (2, 3) and dynamics governed by equations (5, 7, 9), are called Friedmann-Lemaître universes (‘FL’). The Friedmann equation (9) controls the expansion of the universe, and the conservation equation (5) controls the density of matter as the universe expands; when S ̇≠ 0 , equation (7) will necessarily hold if (5, 9) are both satisfied. Given a determinate matter description (specifying the equation of state w = w(μ, T) explicitly or implicitly) for each matter component, the existence and uniqueness of solutions follows both for a single matter component and for a combination of different kinds of matter, for example μ = μbar + μrad + μcdm + μν where we include cold dark matter (cdm) and neutrinos (ν). Initial data for such solutions at an arbitrary time t0 (eg. today) consists of,

• The Hubble constant H0 := (S ̇/S)0 = 100h km/sec/Mpc;

• A dimensionless density parameter Ωi0 := κμi0/3H02 for each type of matter present (labelled by i);

• If Λ ≠ 0, either ΩΛ0 := Λ/3H20, or the dimensionless deceleration parameter q := −(S ̈/S) H−20.

Given the equations of state for the matter, this data then determines a unique solution {S(t), μ(t)}, i.e. a unique corresponding universe history. The total matter density is the sum of the terms Ωi0 for each type of matter present, for example

Ωm0 = Ωbar0 + Ωrad0 + Ωcdm0 + Ων0, —– (10)

and the total density parameter Ω0 is the sum of that for matter and for the cosmological constant:

Ω0 = Ωm0 + ΩΛ0 —– (11)

Evaluating the Raychaudhuri equation (7) at the present time gives an important relation between these parameters: when the pressure term p/c2 can be ignored relative to the matter term μ (as is plausible at the present time, and assuming we represent ‘dark energy’ as a cosmological constant.),

q0 = 1/2 Ωm0 − ΩΛ0 —– (12)

This shows that a cosmological constant Λ can cause an acceleration (negative q0); if it vanishes, the expression simplifies: Λ = 0 ⇒ q = 1 Ωm0, showing how matter causes a deceleration of the universe. Evaluating the Friedmann equation (9) at the time t0, the spatial curvature is
K0:= k/S02 = H020 − 1) —– (13)
The value Ω0 = 1 corresponds to spatially flat universes (K0 = 0), separating models with positive spatial curvature (Ω0 > 1 ⇔ K0 > 0) from those with negative spatial curvature (Ω0 < 1 ⇔ K0 < 0).
The FL models are the standard models of modern cosmology, surprisingly effective in view of their extreme geometrical simplicity. One of their great strengths is their explanatory role in terms of making explicit the way the local gravitational effect of matter and radiation determines the evolution of the universe as a whole, this in turn forming the dynamic background for local physics (including the evolution of the matter and radiation).

Spirit is Matter on the Seventh Plane; Matter is Spirit – on the Lowest Point of its Cyclic Activity; and Both — are MAYA. Note Quote.

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In the 1930s the scientist Sir James Jeans wrote:

the tendency of modem physics is to resolve the whole material universe into waves, and nothing but waves. These waves are of two kinds: bottled-up waves, which we call matter, and unbottled waves, which we call radiation or light. If annihilation of matter occurs, the process is merely that of unbottling imprisoned wave-energy and setting it free to travel through space. These concepts reduce the whole universe to a world of light, potential or existent . . . . — The Mysterious Universe

The idea of matter being crystallized light echoes what H. P. Blavatsky wrote half a century earlier in The Secret Doctrine, where she speaks of “that infinite Ocean of Light, whose one pole is pure Spirit lost in the absoluteness of Non-Being, and the other, the matter in which it condenses, crystallizing into a more and more gross type as it descends into manifestation” (The Secret Doctrine). Material particles, she said, were infinitely divisible centers of force, and matter could therefore exist in infinitely varying degrees of density. Our physical senses have been evolved to perceive only one particular plane of matter, which is interpenetrated by countless other worlds or planes invisible to us because composed of ranges of energy-substance both finer and grosser than our own.

Modern science has analyzed matter down to the point where it vanishes into wisps of energy. Energy is said to be a measure of motion or activity. But motion of what? It is a truism that there can be no motion without something that moves. Scientists in the last century believed that wave-motion took place in a universal medium called the ether. This hypothesis was abandoned because the ether proved to be chemically and physically undetectable, and science was left with the unlikely idea that waves are transmitted through “empty space.”

Modern physicists believe that underlying the material world there is a quantum field, also called the quantum void or vacuum. The quantum field is said to be “a continuous medium which is present everywhere in space” (The Tao of Physics) and matter is said to be constituted by regions of space in which the field is extremely intense. Scientists assert that the quantum field is non-material, but deny that it is mere nothingness. Paul Davies states that the quantum void is not inert and featureless but throbbing with energy and vitality, a seething ferment of “Virtual” particles and “ghost” particles. (Superforce) It therefore seems to be actually a form of ether, which is non-material only in the sense that it is not composed of physical matter. Rather than material particles being “knots of nothingness,” as Davies calls them, they may therefore be seen as vibrations in an etheric medium composed of a subtler, superphysical grade of substance. The same reasoning applies to all the other “non-material” fields and forces postulated by science.

Everything is relative. Physical matter is condensed energy, but what for us is energy would be matter for beings on a higher plane than ours, as is suggested by the fact that energy does not exist in a continuous flow but is composed of discrete units or quanta. Likewise, the energy on the next plane would be matter to an even higher plane. The loftiest form of energy in any particular hierarchy of worlds is what we call spirit or consciousness. As H. P. Blavatsky put it: “Spirit is matter on the seventh plane; matter is Spirit – on the lowest point of its cyclic activity; and both — are MAYA.” (The Secret Doctrine). To say that spirit and matter are “maya” or illusion does not mean that they do not exist, but that we do not understand them as they really are. Any particular plane of energy-substance can be understood only with reference to superior, causal planes. Everything — from atom to human, from star to universe — is the expression of something higher.

Throughout the ages, sages and seers have suggested that hidden within the phenomenal world in which we live there are inner worlds of reality — astral, mental, and spiritual — and that the physical world is but a pale shadow of the spiritual world. These inner worlds cannot be investigated with physical instruments, but only by delving into the depths of our own minds and consciousness, and this requires many lives of self-purification and self-conquest. Scientists using only materialistic methods are in no position to deny point-blank the possibility of such higher planes.

Most scientists, in fact, now believe that some 90% of the matter in the universe exists in a state unknown to them; it is called “dark matter” because it is physically unobservable, and its existence is known of only by its gravitational effects. Such matter is suggestive of the higher subplanes and planes postulated by theosophy, which are composed of matter of increasingly slower rates of vibration and are therefore beyond our range of perception. Given scientists’ confessed ignorance of most of the matter in the universe and their inability to explain satisfactorily the evolution of life and consciousness and the “laws of nature” along materialistic lines, any suggestion that they are on the verge of discovering the innermost secrets of nature or of reducing the mystery of existence to a single equation is premature to say the least!

In theosophical philosophy, the physical universe is regarded as no more than a cross section through infinitude. Universal nature is composed of worlds within worlds within worlds, filled full of conscious, living beings at infinitely varying stages of their evolutionary awakenment. Our finite minds cannot embrace the infinite. As G. de Purucker says in his Fundamentals of the Esoteric Philosophy, we can do no more than to try and form a simple conception of the Boundless All: never-ending life and consciousness in unceasing motion everywhere. The ancients, he says, were never so foolish as to try to fathom infinitude. They recognized the reality of being and let it go at that, knowing that an ever-expanding consciousness and an ever-growing understanding of existence is all that we can ever attain to during our eternal evolutionary journey through the fields of infinitude.