The Lion in the Aqueduct

Rushing by, washing away, deepening down, closing off,
Each droplet flowing to the ocean.
The walls crumble to sand; the sand gets carried away
Even as we do.

As we bring our boundaries down, we scrape too at the floor:
Walls once sand are sand no more -
Digging, digging, we find walls of sandstone.
We build the aqueducts that will form our world.

Water only rushes where water flows,
Boats can only navigate where they run not aground.
They call us the water bearers, but the water bears us.
They say, fear not the apotheosis of the ocean,
For even as clouds bear droplets to mountaintops,
Our river is fueled by those who lose themselves in the only dance.

Never, roars the Lion.
River shortens, river widens, and all is lost in the morass.
The soup of tides spreads each day as the river carries away its bounds.
Tear down that wall, it was said, so instead we live in a trench.

It is inevitable that rivers wear at their banks,
That we avert from leaning over the edge of canyons,
That mountain climbers are seen as just a little bit crazy.

Just as it is inevitable that we yearn again for the idyllic.
Would we recognize it if we saw it?
Are our eyes sealed with the sand we've scraped away?

Some who don a mane bleat that the end is near,
That we will be the victims of population control,
Engineered pandemics, sinking continents, solar flares, meteorites.

But there is no emancipation, only waiting.
The chreode becomes so stable it collapses.
Nothing escapes, not even light, all trajectories point home.
As above, so below.

But no matter how we try, we can never drown the Lion;
The Lion grows gills and will sting those who would try.
The tamer who lets his guard down is maimed,
And tension is the key to instability.

We must trust in inevitability.

Copyright J.K. Strain, 2013.

Frames and Elements; Divergence and Convergence

Just before I hopped in the shower today, I picked up Carse's Finite and Infinite Games and reread a couple passages.  In my esteem, James Carse deserves cannonization (sic.) as a 2nd-Class Saint of Eris, given that said book boils down to a case for the idea that all seriousness is contained within playfulness, rather than the other way around, and given the way his delightful Religious Case Against Belief points out that the complete acceptance of any literal religion as a guide appears to be an abdication of moral decision making in favor of letting some text decide for you.

Delightful as I find the books themselves, I'd rather talk about the train of thought the former set off in me.  Reading the book I remembered the period soon after I'd first read it before finishing undergrad classes, and some of the discussions I had with fellow philosophy students.  Of course, contention comes to memory more easily than quiescence, so my mind quickly drew back to a peer's point about the great difficulty in defining 'game' as Carse uses it in the book.  Perhaps due to our shortsightedness, due to the lack of a cogent phrase, or simply due to misconceptions about what kind of definition would satisfy, we had no success, with effort, at defining the term in a way that seemed satisfactory.  All the definitions proposed seemed either too limited in scope to fit the way the term comes up in the book, or so broad that we could see nearly anything in terms of games.

It was today that it dawned on me that that the latter doesn't actually a pose problem.  If a term's definition has enough flexibility that the term can be applied to virtually anything, the term doesn't refer to an element of the world, it refers to way of framing our observations of the world.

Previously I've heard it argued that terms that can apply to anything, in fact, apply to nothing.  This point holds water in that, if you treat everything as though it had a particular property, explanations in terms of that property tell you almost nothing.  For instance, if we posited that the universe and all constituents thereof are 'wiggly', we could attempt to explain both the counter-intuitive results of the double-slit experiment and the poor record of intelligence tests at predicting future performance in terms of wiggles, but to the extent that we posited wiggles as explaining every nook and cranny we could project them into, we would simply create a semantic spook, like the imps medieval doctors posited to reside in the abdomen that caused stomach cramps.  Similarly, atheists (and others) will readily note that explaining the unknown in terms of "God did it" and leaving it at that serves more as a veil than an explanation.

To this I offer the rejoinder that such explanations attempt to explain happenings in the universe in terms of some particular element of the universe (or, to preempt the quibbles of certain theologies, in terms of some element that has causal efficacy over the universe).  We posit wiggles, God, or phlogiston as referent terms rather than as frames of reference.  Games, however, as presented in the book, feel more like a way of looking at things than a set of elements that explain the operation of other elements.  (It's worth noting that Carse does not make this distinction explicitly, but we challenge you to tell us how an infinite game would operate without multiple cognitive frames of reference.)

Now let's spend some time looking at the inherent barrier that sometimes hides the distinction between frames and elements from us.  The primary one, as I see it, lies in the dominance of convergent thinking.  If the distinction between convergent and divergent thinking doesn't ring a bell,  convergent thinking involves the attempt to find one solution to fit all the evidence and moves onto a different once such a solution has been found; while divergent thinking entails finding many possible solutions to explain extant evidence, with more of a focus on creating possible answers than on eliminating possible answers to find the solution.

With convergent thinking, novel observations lead us to either find a more accurate way to describe the behavior of currently accepted elements of the world or to posit new elements such that their interaction with previously known elements explains the new phenomenon.  Convergent thinking aims to create a mono-mega-model (MMM) for explaining everything smaller than the universe and larger than the Planck scale, so developing new models without a way to tie their framework into that of the MMM fragments our ability to understand the universe.  The urge to convergent thinking gives the appearance of a crisis in modern physics, with the macrocosm explained in terms of the bending of continuous spacetime and the microcosm in terms of discrete quantized units of the same.  We create new frameworks, but only with the intent of unifying what we previously dealt with in multiple different frames.

As David Bohm points out in Wholeness and the Implicate Order, chapter 5, each major tangible advance of science, each paradigm shift, indicates itself by the explanation of what were previously dealt with under different frameworks in terms of a unified order.  Bohm, though hardly the exemplar of convergent thinking's worst shortcomings, still indicates some of them in his thought.  He makes a strong stand in the first chapter against the widespread assumption our theories give us a complete and perfect picture of reality, checking the tendency of convergent thinking to lead to an MMM we believe in because we have nothing else, but he makes a very determined case against fragmentation.  Though not opposed to divergent thinking,  it seems pretty clear that he feels convergent thinking should act as the final arbiter of our ontology.

On the other hand, modern divergent thinking typically involves at least the tacit acceptance of confirmation bias as remarkably difficult to eliminate from thought - so difficult to eliminate, in fact, that you easily to think you've eliminated it when, in fact, you haven't.  With divergent thinking one prefers to have multiple frames that explain the same phenomena than to narrow down to one on a permanent basis.  This doesn't necessitate that the divergent thinker freezes when it comes to choosing a course of action where different frames point in different directions (though premature uncertainty poses as much a pitfall for divergent thinking as premature certainty does for convergent) - it simply involves coming up with multiple solutions to the same problem before riding off into the sunset on the back of the first seemingly cogent explanation.

Studies have shown that the brainwaves of creative women performing divergent thinking tasks tend to be asynchronous, while while those of creative men doing the same are more often coherent across the cerebral hemispheres.  Given [pdf warning] that a significant portion more left-handed people (more often those who write with their hand above the text than those who write with their hand below it as most right-handed people do) have language centers outside the left hemisphere than do right-handed people, this indicates why left-handed men do markedly better on average than right-handed men do at divergent thinking tasks, while handedness does not appear to affect the performance of divergent thinking tasks for women.  It also indicates an explanation for the cultural dominance of convergent thinking.

While the influence of women among intellectuals has markedly increased over the past several centuries, and left-handed males have an influence on politics and the like disproportionate to the amount of the population they constitute, by millennia of patriarchy and by sheer numbers, right-handed males, least likely to be particularly gifted at divergent thinking, have constituted the majority of influential positions in modern history.  Hardly a surprise, then, that we often tacitly accept convergent thinking as the final arbiter of thought.

Now that we've examined the reason we tend to use ubiquitous terms (terms that can be applied to nearly everything, remember) as elements rather than frames, let's time to look at the two with an eye toward recursion.  Of course, in the frame we've looked into here, we see frame and element as the two major elements.  Distinguishing between referents and frames of reference comes in handy when looking at perspectives, but it is still but a way of looking.  Seen through its own eye, it assumes that all ways of looking have assumptions underlying them so that it can compare those, and places perspectives in terms of the frame that created them rather than their content.

This arrangement reminds me of the epistemic paradox underlying much of post-modernism: why trust the frame that tells us no frame is privileged over various frames which claim that they are privileged?  As tempting as we find it to say that any frame which makes an idol of itself sells itself short when it comes to seeing the whole truth, enacting this belief idolizes frames that don't affirm their own truth via circular logic.  Such seeking to winnow away frames seems to bear the mark of our favoring of convergent thinking over divergent.  If we don't expect to find singular truth for the sake of accommodating our habits of thought, little issue arises with holding many frames in approximately equal esteem.

In the opposite direction, taking frames as elements, we find a frame that shows us various ways of interpreting perception as elements acting upon the sphere of human action.  Such a frame would resemble the one used in the above discussion of convergent and divergent thinking as elements affecting human thought.  Such a frame has much use in times where many competing perspectives affect people's behavior.

The status of cosmology and the place of certainty in it

Since I figure a fair portion of you guys are interested in science, today I'm going to write something about cosmology.

One of the issues that faces cosmology and, to a lesser degree, astronomy, that the "hard sciences" like chemistry and some parts of physics don't have to deal with is the difficulty of performing experiments.  Unless we physically send a space probe or rover, capable of performing experiments and sending back data to us, to whatever reach of space we're curious about, we only have observation to go on.

This is why we operated for over a millennium on the idea that Sol and the planets orbited Earth.  When we saw retrograde motion ( http://www.lasalle.edu/~smithsc/Astronomy/retrograd.html - little need to read much of the article, just look at the animations if you're not familiar with it and you'll get it quickly), we decided that the planets moved around the Earth in cycles within cycles (as shown in the second animation on the foregoing link), called 'epicycles'.  However, more precise observation showed that this could not account fully for the motion of the planets through the constellations, so enterprising astronomers added cycles within epicycles.  The predictions the second-order epicycle model were also found wanting for predicting observation, so third-order epicycles were added, and so on.  This could have gone on forever, nesting ever more cycles within cycles, and with computer programs of the type that create fractals, we might have modeled the position of the planets in the sky to an arbitrary degree of precision.  (We would have had more difficulty modeling their distance, but we didn't know how to measure that back when we used the Earth-centric model.)

However, before we got computers, Copernicus came along and dredged up the old Greek model of planetary motion, where the planets, Earth included, all orbited Sol.  Other astronomers gathered data on the planets' angular positions which would support a modified form of Copernicus' hypothesis, but it was only with Galileo that a powerfully convincing case was made that the planets all orbit Sol.  If the data matched one model basically as well as the other, why bother changing the tried and (basically) true methods of calculating the planets' positions?  But Galileo had the telescope, and with it he was able to overturn the main reason for believing in an Earth-centered cosmos: with his new tool of observation, he showed that Jupiter had moons orbiting it.  If some things had been shown not to orbit the Earth, there was no longer reason to assume that everything orbited the Earth.  Consequently, the Sol-centered model won out and now, 500 years later, some educated people still tacitly consider the Earth as the center of the universe, but when they think about it actively just about every educated person believes consciously if not unconsciously that we're on an approximate sphere that hurtles around Sol once every year.

(As an aside, it's interesting to note that under Galilean relativity of motion, we could make predictions just as correctly if we thought of Sol and Luna as orbiting the Earth, and the other planets orbiting Sol.  With motion, any center location is as accurate as any other, but some are more convenient for thinking about certain types of things.)

Now, the reason I bring up this brief history lesson is to look at what it took to change paradigms.  After Copernicus, some astronomers used the Earth-centered model for describing the positions of the planets, and others used the Sol-centered model.  Neither of them disagreed very much in the observed data that their models predicted, but the method used by the two models to arrive at those predictions differed starkly.  Now that we've had the telescope for four centuries, and sent out space probes into the solar system, we can say pretty definitively that the model where all the planets orbited the Earth in epicycles was flat-out wrong, and the only reason it made predictions that agreed with observation roughly as well as the Sol-centered model did was that it had centuries to accommodate itself to observation by adding more and more layers of epicycles.  There wasn't really a competing theory to stomp it out, so it had plenty of time to adapt to its environment (Sol-centered theory at the time still used circular orbits, so the angular predictions of the Earth-centered model were actually more accurate the first time Sol-centered theory died out).  Of course, once the environment it had adapted to changed with the ability to observe the sky in a new way, it lost out to its recently rejuvenated competitor.

Another thing that's worth looking at about this is the difference between the mindset that comes from believing everything in the cosmos orbits the Earth and that one that goes along with believing you're on an orb orbiting something else.  With the perspective that cosmos revolves around your world, it's very much easier to believe in a deity who specifically had humankind in mind when creating the cosmos, but more importantly it's easy to take human assumptions and perceptions about the world as the world itself.  With a Sol-centered perspective, the foibles of humanity seem not so much less significant as less universal.  With the resurgence of the Sol-centered perspective came the widespread use of the scientific method, in which one tries to divorce one's ideas about the world from the human perspective to apprehend objective truth.  With it has also come, after more time and further scientific discovery, the assumption among characters like Stephen Hawking that all human experience is an illusion created by the brain for the purposes of evolutionary survival.  Others argue that treating mind as an exclusive property of humans is another form of Earth-centered thinking trying to sneak back in.  At any rate, the transition to the Sol-centered perspective helped us tremendously insofar as developing technology at a remarkable speed, making a much larger portion of the population educated, and providing food for a greater portion of the human population is helpful.

Outside of astronomy, ideas about the cosmos provide a context for human lives.  Thus, when we have multiple models that describe the same data, we have another criteria that lets us choose which to use for the moment: not only can we decide in terms of which one currently holds the minds of the scientific consensus or which one makes makes fewer major assumptions (deciding via Occam's razor), we can also decide in terms of which one provides a context for human lives that is more helpful for our survival as a species.  These are all in absence of a new groundbreaking observation that lets us rule out some theories.  To put it in Ramsey Dukes' terms, when it comes to selecting ideas about the cosmos, there's the purely Scientific method where a particularly groundbreaking observation rules out some models, but within the remaining models we have the Religious method where we operate under the authority of scientific convention, the Artistic method where we pick the remaining model that has the most parsimonious structure, and the Magical method where we chose from the models Science has not ruled out in terms of which would make the useful beliefs easy to access.  (Remember that these decisions are only about models that observation has not ruled out - no matter how helpful or aesthetic it might be, people's experience with knots and other three dimensional objects would make it difficult to convince anyone that reality was actually two-dimensional.)

The reason that Science and the experimental method is more limited in the case of cosmology and must rely on Religion, Art, and Magic (note the capitalization - those terms mean a specific thing here as defined in the last paragraph) between discoveries of new tools of observation is that we can't actually do experiments with most of the things we hypothesize about in cosmology.  Whereas with particle physics, we can observe, hypothesize, experiment, analyze our data, and form conclusions, with cosmology we are can only observe, hypothesize, analyze the data in terms of our hypotheses, and form tentative conclusions.  The only way we can use experimentation to further cosmology is to extrapolate testable predictions from cosmological models, or develop tools of observation that might undermine the assumptions of certain models.  So we can only use Science in some cases.

With the Religious method of selecting models, we put stock in whatever model most scientists, or the most authoritative scientists, have put their bets on.  This is useful for forming a common framework for studying something, but it carries the problem of group-think and artificially maintained consensus with it.  Consider the story of the ethologists who put ten chimps in a room with a banana hanging from a rope with a ladder under it.  One of the chimps climbed the ladder to reach the banana and the experimenters electrified the floor, so the other chimps beat the crap out of him.  They took a chimp out at random and replaced him with another, and of course he immediately made his way up the ladder.  Again the scientists electrified the floor, and the other chimps attacked him.  After a couple more times of this procedure, the scientists didn't need to electrify the floor to get the old apes to attack the new one who tried to climb the ladder for the banana.  They scientists continued replacing apes after they stopped electrifying the floor, and even after all of the apes who had been electrified when their new fellow had tried to steal fire from the gods were gone, they still attacked new chimps who tried to climb the ladder.  All of the old apes had been attacked when they tried to climb the ladder, so they attacked newcomers who tried to do the same.  Deciding models purely based on consensus carries with it the issue that old, incorrect models might have incorporated ad hoc hypotheses (like cycles-within-cycles) over time to make themselves impervious to refutation by existing observation.  If it's simply convention to lynch new chimps who try to climb the ladder, it means little to say that the floor doesn't get electrified anymore.  More importantly than all that, though, at some point you can trace any consensus model back to when it was new.
So models can be maintained by the Religious method of model choice, but no model becomes the consensus by the Religious method.

Since all Scientific observations can be explained by multiple models (particularly with cosmology), and Religious consensus cannot explain why any model among many compatible with observation became consensus in the first place, we must trace all models back to their Artistic and Magical import.  Occam's razor, the idea that we should not multiply explanatory entities more than necessary, is one of the basic Artistic principles for model selection.  After it was realized by Maxwell that light propagates at a basically fixed speed as a changing magnetic field causing a changing electric field causing a changing magnetic field and so on, we could have added some ad hoc hypotheses to the idea of the Luminiferous Ether (the fluid through which electric and magnetic fields propagate as waves in 19th century models of electricity and magnetism) but it was far simpler to say that the fields simply propagate through space.  So the Artistic method is a good way of getting rid of old hypotheses that are clinging on by a tenuous mesh of complication.  Sometimes, however, it's difficult to assess whether one theory is more complex than another, particularly by scientists who are trained as experimenters rather than as logicians, so the Artistic method is most useful for telling us when observation has significantly ruled out a model, even though it has ad hoc hypotheses to justify why it was actually right all along, but just in a different form.

There have been few paradigm shifts made on Magical basis, but every paradigm shift has had Magical consequences.  Germ theory, for instance, brought us a set of oblations which we perform before eating and, more importantly, before surgery, with which we consecrate our food and make gestures to ward off malign spirits when we sneeze.  Similarly, Behaviorism - the idea, developed through experiments where organisms were treated as black boxes with only input and output, that systems of punishment and reward can teach any behavior to an organism that is biologically capable of it - had a profound impact on the public schooling system and child rearing.  Many people use a Magical basis to decide which interpretation of Quantum Mechanics to use.  Since there are many interpretations of QM which agree with the data, this is completely justifiable.  (Of course some people take these interpretations and forget that they are derived from properties that operate at the minuscule scale of the quantum world, writing books like the Secret which are pure wish fulfillment and have lost all scientific basis, but have great sales figures.  If consciousness can influence quantum indeterminacy, it would have to influence chaotic systems for its small-scale changes to have any noteworthy effects.)

Models of the fabric of the universe like the interpretations of Quantum Mechanics have a profound influence on our ideas about what we are and what we're made of, but cosmological models are at least as important - they affect the way we put our lives in context, and our ideas about "what it all comes down to."  This is seen pretty profoundly if you contemplate for a moment that we're talking assemblages of bio-goop on a speck orbiting a spark in a corner of the universe that seems rather ordinary - a pretty good way to make everything in your life seem insignificant for a moment if things seem to stressful, but extended contemplation can lead to profound existential crises far beyond the scope of this discussion.  And that's only the individual contemplation of an idea - when a nation or culture takes up a cosmological model in exclusion of others, that model is sure to have profound if subtle influences seen in the effects of the behavior of the people in that group.

Currently, the consensus model of cosmology is Big Bang theory.  (Illustration: http://imgur.com/TlmWD )  For astronomers, in its most basic form it simply means interpreting the fact that we don't see a star at every possible point in the night sky in terms of the fact that the speed of light is finite and the idea that the universe has not been around long enough for light to get from everywhere to everywhere, and interpreting the fact that light from distant galaxies has a longer wavelength than light from closer galaxies in terms of the idea that the universe is expanding and cooling.  For cosmologists, popular science writers, and the general public, it means that the universe came into being a particular duration of time ago, and has been expanding ever since, and will eventually either start contracting together until it smashes back into nothingness (the Big Crunch is less popular among scientists than it used to be) or it will eventually expand so much that information, heat, and energy will become basically meaningless.  Heat Death is the favored end foreseen by cosmologists, it's not really surprising that our predictions of the cosmos' future and our rapid consumption of resources with little regard to sustainability reflect one another.

Currently still somewhat close to the fringes but gaining momentum are various hyperspherical models of the universe.  These models describe an ageless universe with no beginning or end, where gravity exactly balances entropy, and life planets of infinite variety exist every now and then stretching beginninglessly back and endlessly on through time as stars are born and die, galaxies form and collapse, and the cosmos carries on.  A hypersphere is the simplest four-dimensional shape, and since Einstein's principle of invariance we've known that all our calculations about constant speeds come out correctly if we treat time as the fourth dimension, perpendicular to the other three similarly to the way the three spatial dimensions are perpendicular to each other.  A hypersphere has no outside which can be reached by entities inside it, but it also contains no boundary - just as you can go as far as possible on a globe in a straight line and get back to where you started, traveling twice the diameter of a hypersphere will bring you back to your starting point, and just like all straight-line paths you might take around a globe meet at both the point where you reside and the point on exactly the opposite side of the sphere, every location in a hypersphere has an antipode exactly one diameter-length from it in every direction.

[The simplest way to imagine a hypersphere is to think of two spheres whose surfaces touch.  Not only do they touch at one point, but every point of their surfaces correspond with each other.  If you draw two concentric circles, imagining the inner circle as the first sphere, the center of that circle is the center of the first sphere.  The outer edge of the outer circle is the center point (compress it in your mind) of the second sphere, which comprises the part of the outer circle that isn't covered by the first sphere.  You can think of it as two spheres next to each other, rolling to touch wherever you would cross between them.  Of course, wherever you are on a hypersphere seems like the center of one of the spheres.]

In a hypersphere cosmos, the reason that we don't see stars in every possible place in the night sky is not because the stars haven't been around long enough, but because the cosmos is curved and we can't see past its horizon.  The lengthening of light from distant galaxies happens because their light lenses through the curvature of the hypersurface (as shown here: http://imgur.com/YCL0q ) in the hypersphere model.  In some versions of the hypersphere model, the cosmic microwave background radiation, which had to be filtered to remove the influence of nearby bodies before we got the most common picture of it, is gravitational radiation emitted in real time rather than something that a newborn universe made only at a certain stage in the process of its cooling.  The fact that light lengthens at an increasing rate with distance rather than the linear one you'd expect with a uniformly expanding universe (explained in terms of negative pressure and accelerating expansion - "dark energy" as it's called - in the expanding cosmos model) is explained in terms of the fact that the the greater the distance between two points, the greater the lensing effect between them (lensing is added cumulatively since each point on the hypersurface bends the light, giving the increasing rate of lengthening with distance).

With small modifications, a hypersphere theory can be created that models most of the observations that big bang theory can be modified to model.  Since the core difference between them lies in the interpretation of redshift, and subsequent differences result from the fact that one model looks to explain things in terms of the universe having a beginning and the other looks to explain things in terms of processes that can be carried out indefinitely in an ageless universe, and since cosmologists can't do experiments (caveat: there's at least one hypersphere theory that predicts novel and minute variations in the gravitation of different isotopes based on their nuclear structure rather than their nuclear mass, but which has not as of yet been tested due to lack of resources), we're left with a dilemma at least until Science comes along to save us.

The Religious consensus, of course, favors big bang theory, and will pull out tons of data interpreted in terms of big bang theory as evidence that it is correct.  Being that it's the consensus, it has the manpower to interpret more data in terms of big bang theory than competing theories have to interpret data in their own terms, so the presentation of evidence interpreted in terms of the consensus theory without creating contradictions falls a fair bit short of proof.  This evidence is of course still worth consideration, but quantity should not be a conclusive decision point, at least until we reach a point where the two sets of models are on equal footing in terms of manpower and resources for the type of Scientific tests mentioned parenthetically in the previous paragraph.

Likewise, the Artistic case is not conclusive.  Hypersphere theory has on its side the fact that an ageless universe where time is relative makes more sense than a universe with an absolute age where time is relative, as well as the fact that the creation of a transfinite amount of mass-energy to fill a transfinite and expanding amount of space-time with relative uniformity would require an exception to the conservation of energy.  Dark energy seems to fall under Occam's razor in some portrayals, but in a flat universe it has the beauty of perfectly balancing the force of gravity (if the universe is flat after all).  Hyperspheric self-containment is simpler than an unbounded infinite quantity of space, but a set antipode length is just as arbitrary as a set age for the universe unless it can be defined in terms of fundamental quantities.  Continuous gravitational radiation on the electromagnetic spectrum is not a possibility that is considered in the scientific consensus with the focus of Standard Model on finding a quantized force carrier particle for gravity, but if it is found to be the case in coming years it might make a more parsimonious explanation for the cosmic microwave background radiation.  Finally, if the universe is curved into a hypersphere by virtue of the gravity of its own mass-energy as some models suggest, this might lead to both the acceleration experienced in the Pioneer anomaly and the unexpectedly fast spinning of galaxies that, decades ago, led scientists to postulate that a tremendous amount of undetected dark matter permeated galaxies and caused the increased angular speeds.  Without further research, the Artistic case for the hyperspheric model is not conclusive, and without funds allocated to research that would test some of these possibilities, it is unlikely to become conclusive.

The Magical case is somewhat more conclusive, though there are still arguments to be made on each side.  Big bang theory has the fact that a universe which has a beginning is much easier for humans to understand on its side, and with it the hope that a fairly complete history of the universe may eventually be developed.  People with theological reasons to believe in creationism, such as Georges LeMaitre who proposed the theory in the first place, also have a better ally in big bang theory than they do in the hypersphere model.  Since a great portion of the world's population believes the universe was created at some point in the past, this is a good Magical (if not Scientific) reason for exploring the possibilities of big bang theory fully before leaving it behind if we leave it behind at all.  A universe with finite age also gives humans a greater cosmic importance in a way, since we are living in one of the most interesting times in the life of the universe.  (It does not necessarily confer us special status as one of a finite number of sapient species however, given that space is infinite with relatively uniform mass-energy distribution in modern big bang theory.)  If time goes on endlessly and uniformly, everywhere in the universe will have life at some point, over and over again in different permutations, making our status less special in comparison.

The hypersphere model has some points on its side as well.  For one thing, the image of an intergalactic civilization having conquered so much of the universe but amounting to naught in the heat death of the universe is lifted from our mind's eye.  This gives us incentive to pursue space exploration for its long-lasting fruits rather than with the rush we'd be in under big bang theory.  Rather than rushing out to take what we could and conquer space, belief in an eternal universe would guide us to cultivate the universe like a garden rather than farm it until its nutrients were exhausted.  The same perspective change would apply to our use of Earth's resources - we'd have a reason to preserve life while continuing to cultivate technology for expansion into space, and less incentive to develop technology at a breakneck pace in a rush to find a way to spread ourselves or our seeds to the stars before we destroy ourselves with the weapons made from that technology, or in the scarcity that results from a manic race toward further technology.  With the emancipatory apocalypses written into the dominant monotheist paradigms and the nihilistic "take it before it's gone" written into the dominant materialist paradigm that atheists and other secular types turn to, an outlook like this is desperately needed.  Further, if big bang theory has on its side that things with beginnings are familiar to humans, there's something intuitively natural about circles, spheres, and n-spheres for us as well.  Besides those created by our limitations (either limitations of our capability or limitations we place on ourselves), we never encounter boundaries in the world, but horizons are quite familiar to us.

While big bang theory, besides the familiarities it holds, offers a sense of urgency to no ultimate purpose (in the currently triumphant heat death version of the theory particularly, but the big crunch is little better), the hypersphere model holds sustainability and the opportunity for as much continuity as reproducing living beings could ever want.  It entices us into space with the possibility of infinite play rather than sending us out to claim spoils, and it heals us of some of the desperation of being alive in one of the only times that life is possible while being so silly as a species.  It also alleviates the sense that without humanity expanding to the universe and beyond, the universe will be a meaningless non sequitur.  Rather than pushing with the cruel hand of scarcity of resources and time to enjoy them, it pulls us back, urging us to slow down and enjoy our development rather than rushing to destroy each other before it's all gone, guiding us to be more playful with our existences.  After all, who wants an eternity of seriousness?

To be clear, I'm not suggesting that we abandon big bang theory to solely pursue hypersphere models at this juncture.  I only suggest that, in the absence of observations that irrevocably remove one or the other models (or both, wouldn't that be a wild surprise!) from the realm of possibility, we make our personal decisions with the possibility in mind that the universe could continue ever onward, and that we share our interest in the hypersphere possibility, if we find such interest within ourselves.  Remember, cosmology is not hard science in the same way that chemistry is hard science - we cannot do direct experiments, so a more zetetic multi-model approach is better suited to the subject than a monolithic one.

The Enochian Alphabet: An Interpretation as Ideograms

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The Enochian alphabet captures the interest of many magicians, and though it is sometimes treated simply as the way one writes words in the Enochian language, some have sought to explore its own internal properties, seeking to find in it a magical alphabet akin to the Hebrew alphabet.  (For those not steeped in Western esoterica, a magical alphabet is a set of symbols to which one attributes a variety of metaphorical layers, created for the purposes of taxonomy and for the understanding of words in terms of their letters' symbols.)

The best-known attempt with the Enochian alphabet is that of the Hermetic Order of the Golden Dawn.  The G.D. approach is to attribute the sixteen figures of geomancy and the five elements (the four classical elements plus spirit) to the twenty-one letters.  This approach has the advantage that the elements are quite well-understood in Western occultism, and one can easily come to grips with the geomantic figures through their astrological associations.  This allows one to easily interpret Enochian words as magical formulas (albeit in a magical language that doesn't distinguish verbs and adverbs).  However, the results of this approach seem somewhat arbitrary.  Further, this approach reduces the Enochian alphabet to basically astrological language in the same way that many magicians would balk at if it were done with the Hebrew alphabet.

Even worse, magically speaking, such an approach flies directly in the face of good practice as understood by chaos magicians.  If you're using a magical language which purports to be the first language, you can expect a greater degree of success with it if you treat it as the first language (not because it necessarily is, but because that's how the system was designed).  Rather than interpreting the Enochian alphabet as a particular way of arranging subsequent ideas, the best of modern magical symbology that we can hope to find in Enochian are precursors to more recent systems' contents.  While someone who approaches the issue from a historical perspective may find the fulfillment of their expectations by attributing to the Enochian letters various astrological and alchemical concepts familiar to Dr. John Dee, some of the system's potential for use in magic would be lost in the process.

How, then, are we to find the meanings of the Enochian letters without looking for a later isomorphic system?  With a bit of historical leeway, we can say that the magical alphabet based on Hebrew derives significantly from the meanings of the letters in their pictographic origins.  The letters' names are words which start with those letters, and the letters' symbols are pictographs for those words.  These concepts were used as the basis for the wealth of meanings attributed to the letters - it's quite possible that the astrological symbolism given to the letters in Sepher Yetzirah came later.

However, we can't trace the Enochian letters back to pictography through existing knowledge of them.  Even more relevant, from a magical perspective, the Enochian language was used to create the world, so while the letters may resonate with certain things in the world which particularly express them, their fundamental indications would not be objects.  Instead, an approach to the letters as ideograms is in order.  If successful, such an approach would not only lay bare the words of the Enochian language as formulae, but it would open modern magicians to an entirely new magical alphabet, and perhaps a new way of looking at the structure of the world.

The Enochian alphabet, as first given in Liber Logaeth on page 57.

One easy thing to notice about the Enochian alphabet is that it consists of three sets of seven; this is highlighted in its presentation in Liber Logaeth.  Because of this, we can reasonably expect each letter to share octave-like relationships with two other letters.  We can't, however, insist that these relationships will be in linear order (B going with M going with X, etc) because there may be a hidden structure within the order of the alphabet; such a structure would likely pertain to the Sigillum Dei Aemeth given that both have a septenary structure.

The version of the alphabet used in this exposition is that given in Liber Logaeth, on the basis that the alphabet was first transcribed there, and since (to my knowledge) there is no mention in Dee's journals of the angels revising the alphabet, any alterations in the later forms can be chalked up to transcription errors or intentional changes on the part of humans.  Note that O (the second letter on the bottom row) consists of two lines that almost join but do not, and that K (the second letter on the top row) is one single connected figure, rather than two disjoined figures as it is sometimes displayed.

In what follows, I'll be drawing connotations from these glyphs among the Major Arcana of the Tarot.  This is done specifically with the Atus of Tahuti given in Aleister Crowley's masterpiece in mind.  Rather than to reduce the letters to those cards, the purpose of these connotations is to help cement in the mind the new content given here by connecting it to the known.  The attributions to the Tarot are by no means exhaustive; indeed, these attributions leave a lot of room for personal elaboration.  That said, let us begin.

This letter signifies the /b/ sound.  It resembles the astrological glyph of Aries, and the G.D. attributions treat it as Aries through the tetragram Puer.  It's also interesting to note that the first letter of the Enochian alphabet corresponds phonetically with the first letter in the book of Genesis.  The fact that Aries starts the Zodiac and 'Berashith' begins the book of Genesis reinforces the idea of beginnings already attached to this letter due to its position.  At the bottom, there is a point of bifurcation or joining (depending on which way the glyph flows).  The right branch ends in a concave shape; on the right branch, we have a straight line.  This joining of the phallus and kteis is reminiscent the Greek myth regarding the origin of human love: in an Edenic condition, human beings had two heads, both sets of genitals, four arms, four legs, and so on.  Fearing their power, Zeus (or some other Demiurgic figure) sundered each human in two.  Now humans spend their lives seeking the other half of their souls, yearning to reunite with their perfect complement.  The Hebrew word Elohim is used to refer to divinity many times in the book of Genesis - and it has feminine root with a masculine ending.
Regardless of political agendas and semantic spooks, this is also a glyph of conception as the beginning of life, where the masculine and feminine forces come together.  In the Tarot, Atu VI in its pairing of dualities, I in its hermaphroditic grace, and 0 in its unconditional arising, make good touchstones.  Note that the glyph goes both ways, both as merging and as diverging, as is appropriate for a glyph that indicates the oneness of dualities.

This letter signifies the /k/ sound.  We might say that the wavy line at the right flowers forth from straight the line at the left, or that a line connects the wavy and straight lines together.  The slight bump on the left side of the left line (towards the top-middle) may in fact be an extension of this middle line.  In Alan Watts' terminology, we can think of the straight line as 'prickles' and the wavy line as 'goo' (Watts refers to 'prickles' people, who try to understand things in terms of what they're made of, and 'goo' people, who try to understand things in terms of how they fit into the whole).  So, biologically, we can see a link to the corpus callosum, which links the two hemispheres of the brain.  Turn the glyph 90 degrees anticlockwise and you have a tree, which bridges the ecological cycles underground and above-ground in a similar way.  On the other end of the earlier bifurcation, with the wavy form emanating from the straight line rather than just connecting to it, we have a symbol for pregnancy; in that case the connecting line is the umbilical cord.  More generally, we have a glyph of something not complete in itself, but which can come to completion by sprouting forth: even Πριαπος is not complete until he looses his seed.   In light of the gestation symbol, we can also see the glyph as the present giving forth into a branching future.  Turned around, it serves just as good a symbol of ancestry, of the process by which events converge to yield the present.  Both the past and the future are extensions of ourselves, and we create them using our minds in order to operate as beings not merely in time but of time.  Atu III for the womb, XII for the womb's inhabitant hanging from the umbilical cord, II for the mediation of the divine, and XI for the marriage of Βαβαλον and Χαος.

This letter signifies the /g/ sound, and in some pronunciation schemes, the /dzh/ (English j) sound.  Aside from is very strong resemblance to the letter G in the Latin alphabet, its most obvious feature is probably its spiral shape.  One is reminded of the spiraling ram's horns which, as they grow, maintain the same center of mass due to their expression of the golden proportion.  That proportion occurs everywhere in nature, from the petals of flowers to the shape of the pine cone found on the tip of every thyrsus, to the human skeleton.  It's not only mathematically beautiful, but very sound engineering for structures that grow organically.  Thus, we have the ideas of growth and beauty associated with this letter.  The spiral, infinitely extended, is also perhaps the simplest fractal - at any scale, the same mathematical relations apply.  With the principles of growth and fractality in conjunction, a wonderful modernization of the preformationism implied in י and its corresponding Atu, IX, is found - just as the Hebrew alphabet is generated from י, this letter is the fractal seed which spirals out when planted to bring all under the principles of organism.  That the central seed is buried beneath n layers of spiraling indicates that this order in nature is hidden rather than overt - that the whole visible spiral might, from another perspective, seem just the seed or a part thereof shows how pervasive this order is.  The rounding motion likens this glyph to Atu X (but as cyclic progress rather than cyclic repetition), and the fractality reminds of the as above-so below connection bridged by V: this is the order through which the Hierophant worships.  The organic balance offers VIII a new set of scales, and the order radiates outward in joy like XIX.

This letter signifies the /d/ sound.  It resembles an eclipse in the process of applying or separating.  Thus we see liminality, the suspension of the natural order to which one is accustomed so that transformation can be born.  The glyph also resembles a scarab if turned sideways.  Khepera, the Egyptian scarab god, rolls the Sun underneath the night sky to ready it for the next dawn, and the god's name is closely connected to an Egyptian word for transformation.

(From Wikipedia:)

Kheper-i kheper kheperu, kheper-kuy, m kheper n khepri kheperu m sep tepy.

“I became, and the becoming became. I became by becoming the form of Khepra, god of transformations, who came into being in the First Time. Through me all transformations were enacted.” 

 Turned around the other way, we have the image of a vast pregnant abdomen with the legs spread below it - preparing to give forth with birth.  Easily we can find a common line between the fetal experience of contractions with birth impending (Stanislav Grof's second perinatal matrix) and the apocalyptic sense of experiencing an eclipse.  As transformation, this glyph pertains to those beginnings which are also ends, and vice versa (the middle line is shared between the two semi-circles).  The glyph is also the cauldron, where heat ripens the contents and mixes them by convection.  Atu XIII is nicely encapsulated here, the abdomen of III is featured strongly, and the mixing of XIV.  The shifts implied in the current version of XX are also relevant.

This letter signifies the sound /f/.  It looks a lot like a dragon or an arrow.  It breathes fire upon all and dances in the flames.  Not only is it the primal dragon Tiamat, it is the dragon-turtle Bahamut upon whose back rests the world.  It is Χαος with Βαβαλον astride, and Fafnir lunging forth from his hoard.  As an arrow, its dual crossing lends it quite strongly to the glyph of Sagittarius, The far-reaching intuitive fiery nature of that sign fits well here, though Atu XIV is far less appropriate, unless the cauldron is seen boiling over.The two lines also fit with the birth symbolism from the previous letters, being the cervix and the labia.  Rage, volcanic ecstasy, and Plutonic-Dionysian themes are thus suggested.  Being shoved through a vagina is a very instinctual process, and the will to power leads us as we grow out of successively larger, tightening wombs.  The glyph is very martial, being also a pike or a spear held in two hands.  The pike, used to unseat and impale oncoming horsemen, suggests using the momentum of one's obstacles to overcome them.  Similarly, the child in the birth canal is more being thrust forth than thrusting herself forth.  The spear, first made for hunting, suggests Artemis (and thus Sagittarius again), and of course the arrow springs forth from the tension in the bow, not by being hurled.  The dragon that disdains updrafts wastes energy flying.  The momentum of VII, the primal fury of XI, the implacable inertia of X, the explosion of XX, and the power of IV are all indicated here.

This letter signifies the sound /a/.  In this glyph, we see multiplicity emerging from bifurcation, with paths branching off and converging.  This is the only glyph in the alphabet to contain a cross consisting of straight lines, suggesting an intersection between planes or an orientation (as a compass does).  So we have a world of bending, branching paths which interacts with a more straightforward world (the lower line that goes from up-left to down-right), and the intersection of the two guides us through the branching world.  Sounds exactly like the way that the linear world of perinatal experience helps form our path through life in Grof's ideas, and this orientation process serves a similar role in personality integration as his fourth perinatal matrix.  The paths of the letter resemble the set of all creodes, the totality of nature's habits seen as developmental pathways.  This is the Aether, Yetzirah, the foundation of the material world which contains the infinite set of possibilities for it.  Since it contains all possibilities, it contains no actualities; Yetzirah is to Assiah as Ain Soph Aur is to the ten Sephiroth of the Tree of Life.  So this glyph is the world behind the world, which occultists and the dreaming explore and which m-theorists pray exists.  The G.D. attribution to ו via Amissio is justified through ו's role in the Tetragrammaton as the Son, prince of Yetzirah.  Atu 0 is conveyed by this letter by the fool's position at a precipice, as is the way XIX's sun shines over all, and matrix of possibility in X - though this letter speaks from its axle, not its periphery.

This letter indicates the sound /e/.  Of all the Enochian letters, this is probably the simplest besides the last.  It's a pair of perpendicular line segments joined at the upper right, similar to ד in Hebrew.  It also strongly resembles the previous letter, with the cross and branching formations removed  This limitation to one path indicates that this is a glyph for the material world.  The letter looks like the axes of a two-dimensional Cartesian grid, bringing with it the idea of space.  If, in fractal fashion, we treat each line as the letter itself, we arrive at two pairs of mutually perpendicular axes; i.e.: four dimensions of space-time.  Further understanding of the glyph can be gleaned by treating it as a pathway.  Energy moves in one direction initially, but emerges with none of its push in the initial direction, instead flowing in a way completely orthogonal to its original direction.  Thus, it indicates a transformative function, mediating a force on one plane to a force on a different plane.  Rather than being an extrusion of the last letter, this glyph is both a limited case of the possibilities entailed in the former, and the yoke that allows such cross-sections to be made; that is, it is both what is behind the Gates of Matter and those gates themselves.  It indicates the self-creating material universe that takes inspiration from greater transfinities, Σοφια, the decision made in light of possible outcomes, which allows "outcome" to mean anything at all.  Atu XXI as the World, XVII as a mediator of forces, XV as the Gate of Matter, III as the Door into being, and the rim of the wheel in X.

This letter, first of the second heptad, is pronounced /m/.  Anatomically, it strongly resembles a pair of lips and, simultaneously, a pair of breasts.  The orality and oceanic infant consciousness is thus connotated.  Seen from the side, it also resembles waves.  However, since the glyph also resembles a buttocks, the glyph can't be limited to the oral side of youth.  Instead, it indicates the entire Freudian oral-anal polarity, pitting the shouted voice of authority against the siren's song of comfort-seeking.  Rather than dealing with the sides of this polarity separately, here we have the coin taken as a whole.  The shape of the glyph thus reflects the raising of a child, with two parental curves and a young central point.  With the mammary/gulteal symbolism, we see how the child is nestled/sent forth by the parents.  'Passed down with mother's milk,' of course, is the ancestral memory, the familial unconscious, though to make it the role of one parent alone to pass this on is to demean both.  This letter shows no member of the family more than another, but rather indicates the family unit.  So too it indicates the atom, the fabric of the world just as the family is the fabric of society.  So too, the twin strands of DNA and the path between them that conducts light.  In no case does this letter indicate what is conducted; in all cases it indicates what conducts.  Atu IV, children included, makes a nice representation for this letter, as do the parental pairs - II and IX to birth a god, IV and XVII to birth a lover, V and III to birth an artist.

This glyph is pronounced /i/ or /j/ (that is, the English y) depending on its place in words.  In it, we have what the last letter transmits, passed from the top line to the bottom.  And what a transmission, the power of which brings the whole universe into harmony and resonance! The musician plays, the weaves travel, the eardrum pulses. The eardrum pulses, the receptors transduce, the brain shivers in pleasure. The adherent takes the bread, blesses it, and holds the body. The apopheniac gazes upon the symbol, finds the principle, and sees it instantiated everywhere. Without this power, the universe would be 1 or ∞; with it, it may frolic as naught and two. Without this power, the one could not adore the many, nor the many the one. It is the power of decoding, by which I translate these letters, the power of encoding by which I transmit these words. By this power calls DNA to the protein, choreographing its dance. By this power knows Helium its noble contentment, and finds Hydrogen its consummation in an equal. The fractal order indicated under /g/ would be defunct without its messenger god /i/, and no priest would prophesy the apocalypse of /d/. /e/ creates the field of play for /i/, and together they love as do Nuit and Hadit. No corner of the universe is so remote that the lineage of this letter cannot reach it. It is ubiquitous and interpenetrating , and even as linguistic consciousness and the internal monologue are created by the power of transmission, they are that power. The third I is distinguished in that it doesn't say "me." Atus IX, XIX, XVII, XX (as Heru), and II all traffic with this letter.

 This letter is pronounced /h/.  Just as the stream of water to the ocean creates valleys and caves, so does the power of the previous letter shape what it passes through.  From the inside, the liaison between the arcs is concealed behind the line their union forms, but when the power of this letter is understood, the meeting of the two forces is acknowledged.  Here, apparent opposites make their secret congress.  Thus we have the discreet sex life of the pair in /m/.  Also we have the birth of theory of mind (i.e.: the realization that other people's actions are the result of their own minds), though in others we only have their apparent inputs and outputs.  With theory of mind comes the possibility of deception, just as every shelter is a potential hiding place; with the separate ego comes the possibility of concealment, and vice versa.  Of course, the notion of personal separate existence and theory of mind don't seem to go necessarily in tandem until one notes the secret link between how thought seems from the inside and how it seems from the outside.  Secrecy, of course, is relative - only that which can potentially be known can be secret at all, rather than being simply unknowable.  So this letter pertains to what we hide from ourselves to believe that we understand the world, and to the process of creating perceptual worlds for ourselves through the use of our assumptions.  Belief is the craft of thought, that vast ocean through which we sail.  This glyph signifies the power of the subtle over the manifest.  As sex and the craft of thought, it has ties to Atu VII, also (less strongly, and only the former) to Atu XI, and to XIV.  In the hidden connection of mutually different sets, it pertains to XIII.  As the intermediary between all in the world of duality, it is XV.

This glyph, pronounced /l/, puzzled me for awhile.  In it, we have a binary path encroaching on a circle of undivided substance, or a circle devouring duality, absorbing it into monism.  In either case, this glyph indicates the transformation between duality and unity.  The best anatomical glyph for this process is the optic nerve, whereby the endless light of the world is transduced into something we can make sense of in terms of our perception-conception, which of necessity breaks things into categories.  In that case, it's wort noting that the reality is actually the blank space inside the concave part of the glyph, and the curve itself is the retina, which absorbs this incomprehensibility and communicates something the dualistic mind can make sense of.  Thus, 210 is a numerical articulation of this glyph.  Going the other direction, the formula of ARARITA is disclosed - by pairing each of the ten thousand things with its opposite, we can bring the non-dual truth back into our awareness: All Reflections Are Resolved In The Adept.  There is no bridge shown between monism and nothingness because that bridge is made of nothingness.  This glyph implies VIII in the balancing process, 0 in its indication of nothingness, IV who conquers the world and divides it into boundaries, XII for the immersion in illusion, and especially II.   This letter is a development of the relation between the parts of the previous letter.

This letter indicates /p/.  Here we have the enclosure similar to that of /h/, except that the horizontal lines are at the periphery rather than the center,; what is hidden here is not the convergence of apparent opposites, but the non-identity of an apparent unity.  It indicates an incomplete articulation of the ARARITA expressed in /l/: all observed opposites are indeed reconciled, but this creates a false unity which fails to encompass all - what lies unobserved remains in dichotomy, and because the mark is missed (by taking one's perfectly consistent picture of unity for the world itself), one makes dhyana only upon an idol.  The resemblance between this letter and the Greek Ω is thus made relevant - this letters it an end, but a false end, a dead end.  Here we have shallow holism, that womb-nostalgic perspective which mashes everything together because it doesn't comprehend the order of the universe, rather than fitting it together because ti does.  Holism, of course, is but one of this letter's masks, more generally it is the creation of a purportedly complete picture of the universe based on a small subset of possible data; the trap of this letter is the illusion of completeness that comes with self-contained consistency.  Here lies the 'big' fish who doesn't realize how small his pond is.  This is the letter of false endings, and one of the two deformations of /l/.  Among the Atus, XVI is relevant in that the tower builders took the forces of nature into account, but failed to do so with completeness; graphically it resembles Nuit in XX, who is but a veil for the bornless, and O is also called, in that he wraps himself in a threefold veil such that he knows not where he leaps.

This letter, /q/ (pronounced as a throaty k or as a kw sound depending on idiolect), indicates the second failure of /l/.  While /p/ plunges one into a world of unity propped up by the limitations of any one perspective, /q/ avoids this by a wide stretch: instead of focusing on creating unity for its image of the world, it shapes around itself a world of irreconcilable dichotomy.  Rather than a tent we have a valley, and rather than an enclosure which blends light into shadow and shadow into light, we are cast into the very center of the intercourse between  Yin and Yang (which originally indicated the sunny and shady sides of a mountain).  The world, rather than being a grey place, is a world of qlippothic extremity between light and dark.  Just as /q/ compensates for the debauchery of /p/, so does /p help balance for the decadence of /q/: though the sun shines differently on each side, it is the same fundamental ground of being that makes up the valley.  And in the fractal fashion seen elsewhere in the alphabet, the relation between /p/ and /q/ can be treated in terms of those glyphs.  To /q/ they are most pronouncedly opposite perspectives of which it is unavoidable to take one side or the other (but less obviously they are both ways of organizing perception-conception); to /p/ they are two ways of processing the fundamental unity-dichotomy of reality (to say nothing of its nothingness).  In this regard, the two must be balanced in the harmony of /l/.  Among the Atus, this letter fits nicely with XVIII, that pervasive duality, and with VI as analysis.  In its higher forms it becomes VII, where disparate forces pull in the same direction, and I plays freely in this world of illusion.

This letter is pronounced /n/.  Last of the second heptad, this letter brings the primacy of the last two letters into question.  Showing an arc crossed by an arc, this glyph shows a pair of paths that only cross once, indicating that the two arcs do not enclose an area.  In light of such distinctions, it's clear that all analysis is counterfactual to the extent that it treats the categories it creates as fundamental, and all synthesis is redundant since it's best applied to continent (enclosed) distinctions rather than open ones.  Since the arc does not provide complete division but only partial, this symbol generally applies to the incontinence of apparently real distinctions.  If near death experience accounts are to be believed, this letters is a glyph for dying, as transformation rather than ending.  The various figurative forms of death give us an escape from the traps of /v/ and /q/ into the nonduality-nonunity implied in /l/.  In this defiance of distinction, /n/ is also endless monism rather than the closed monism of /p/; similarly, /n/ contains dualism in its two lines, but they intersect.  Thus, as death/transformation, this letter puts an end to the issues of the second heptad even as it opens the ground for a new transcendent set, even as the perpendicularity of /e/ provided a space for the distinguished polarities of this heptad.  /n/ has ties to XIII for obvious reasons, but the transubstantiation of XVII has its place here too.  XVIII and XIX, together and united, fit as well.

This letter is transcribed as /x/, and presumably pronounced as 'ks' like the English x.  Here we've got a familiar symbol, the perpendicular axis that ended the first heptad, reflected across a vertical line of symmetry (though altered somewhat).  This indicates that the alphabet might be better understood as a reflection across /l/, the middle letter.  Alternately, as this glyph itself indicates, /e/ may be an orthogonal rotation of /x/, with the dot indicating the combination of a particular pair on the axes.  This combination is a refinement of /n/, the last letter: instead of having only one combined pair of points between the axes, all along them potentially combine.  We've moved beyond duality into multiplicity, but 2 remains the door to greater numbers*.  As a result of the pairing, each point on one axis corresponds to every point on the other; thus a surface is enumerated.  The change of order from line to surface casts the previous letter in a new light - it is the equator and meridian of the sphere.  So /n/ remains the symbol of transformation that transcends, but /x/ is the symbol of retroactive transformation, whereby all that has led up to this point is seen as such, rather than as the unfolding of circumstance into who knows what.  A line has no substance - only its angle changes with the shifting of perspective, but a surface can cast as many silhouettes as there are angles of view.  Atu XIV expresses the fullest extent of this integration, VI the loosest.  The mutability over time of XX is pertinent for shifts of perspective; consider also X, which shows different faces from different sides.
*(1*1<1+1, 2*2=2+2, 3*3>3+3, 4*4>4+4, et cetera.  2 is the only number whose addition to itself is equal to its square.)

This letter indicates /o/.  Here we have a pair of arcs (like those in /n/) converging together.  As they get closer together, their trajectories approach one another.  Where the perpendicularity of /x/, /i/, or /e/ suggest the transmutation of one Umwelt or force into another, here we have an approach to fusion suggested.  Astrologically, the resemblance of this glyph to the symbol of the moon emphasizes the idea of undifferentiated communion.  However, when we look closely at the letter as printed in Logaeth (and copied here), it becomes clear that the two arcs do not actually touch.  Thus, rather than a transitional connection between ideas or planes, we have an infinitesimal approximation of two ideas in one another.  Of course, a convergence to infinitesimal approximation implies a divergence to infinite distinction if we simply turn the other way.  In no case however, do these arcs intersect; however, they damn well seem that they should on cursory examination, and in that regard the letter seems the opposite of /q/.  So we have the idea of convergence toward fusion, and the idea that nothing actually touches - it can only come infinitely close.  These two ideas themselves come infinitely close to one another without quite touching, and here we find that fusion is not a noun but a verb; that the experience of fusion with myself is as absent as it is irrelevant and redundant.  I can fuse with an aspect of myself, but only as another aspect abstracted from the whole.  In its highest form, the glyph states the twin truth that love is never complete and love is limitless.  Compare with XV (the illusion of separation), XVII (limitless closeness in infinite space), XX/XII (the inescapable womb) and VIII.

This letter indicates /r/.  Enclosure seems foremost when this letter is contrasted with its cousin in form, /m/.  Where, in /m/, we had a balance of the two arcs, the central merger favoring neither over the other, such is not the case here, as seen both in the downward pitch of the center and the thinness of the top of the letter.  This fits nicely with the concept of enclosure: consider the tendency to perceive the figure over the ground in the West, or vice versa as some cross-cultural psychologists claim occurs in the East.  In eaither case a boundary is drawn, and the attention has difficulty straddling it.  Thus the glyph indicates the ego, which exists separately (or appears to do so) due to the habits of the attention; contrast again with the Nephesh of /m/; so also is it a glyph for the Solar System, to which we occultists generally pay more attention than to the rest of the galaxy (which is, at the end of it, still another enclosure), especially when the membrane of electromagnetic bubbles surrounding it (recently discovered by the Voyager probe) is included in the picture.  So also, on a medium scale, is the atmosphere of our planet an enclosure.  Of course, there are Yetzieraic enclosures at the collective level too, so this letter has some in common with /p/.  It is the process by which paradigms are created, rather than the state of being trapped inside them, once made.  Both are the result of selective attention, but different facets of it.  XIX is quite apt here, but the encompassing arms of III and the lines carved IV's legacy also pertain to this letter.  XXI is relevant to some facets of this symbol, as is V as the Ruach.

This letter indicates /z/.  In contrast to /n/, we seem to have a continent distinction here, continuing the permutations of duality in the second two octaves.  Also like /n/, this letter becomes a set of axes when interpreted in a sphere.  We see a polar axis and a rotational axis, where /n/ showed the directions along the surface.  This spinning explains the previous letter's enclosure as orbit, which is just as valid a metaphor for the containment of attention.  The difference is that in /r/ something other than the bound creates the binding; here the bound is self-binding.  Quantum mystics will see reference to the way quanta can interfere with themselves by creating self-contained virtual quanta.  This glyph, as an indicator of the interface between electric and magnetic fields, generally pertains to the way that internal processes can create external effects, turning spin into thrust.  It makes a nice reversal of the previous glyph in that regard.  Given that the equator and the poles do not touch, the two lines of this glyph are just as disconnected as those of /o/.  Returning to the idea of this glyph as a distinction, we see the type of distinctions that are actually capable of being continent: spin and poles.  The two directions of spin are mutually arising, as are paired poles; thus we see that distinction serves not to separate, but to orient.  Atu XVII, both in its transmutative properties and as a guide to orientation is relevant here; IV, who conquers and around whom others orient themselves and VI, as twin siblings (mutually arising), are also appropriate.

This letter indicates /u/ or /v/, depending on its position relative to vowels and the speaker's idiolect.  Most notable about this letter is that it consists of two similar (if reversed and magnified) arcs, indicating that that which is above is like that which is below, though after a different fashion.  It also indicates the process of bringing into balanced relation the levels of the soul - just as the Nephesh commands G'uph without controlling it, so is Nephesh brought under the yoke of the Ruach, and so on.  The theme of genuine duality is continued, but here we have consonant harmony in the highest forms, and in the lowest, piercing dissonance (the glyph could be basely interpreted as one of figurative or literal pederasty).  Outside the shells, this glyph indicates the continuity that results from the link between levels of reality.  This is the glyph for Knowledge and Conversation of the Holy Guardian Angel, of continuity between the human and the daemon.  The perpendicularity at the point of meeting is illustrative: once the (comparatively) lower arc develops to a certain point, further development is not progress but merely change, sidestepping the genuine issues ahead.  In order for further development to take place, a new approach is needed.  Unfortunately, the new approach must be cultivated before it can even reach as far as the approach which has topped out does, but for progress to occur we can't keep endlessly using the same set of glosses.  Atu V is the most apt symbol here - man as a bridge, reaching up.  XII also relates, reaching downward toward the smaller order.  XIX indicates the arcs of this glyph iterated endlessly.

This letter indicates /s/.  In components, we have a line and a cup, like in /b/, the first letter. The line is horizontal here, indicating, rather than the impetus to ascend, the walking stick with which one treads familiar territory, and the height of the line is a result of the taking-in process of the cup.  Together, the two form a point or the crest of a wave.  In /b/, the two emanate from a point, but here they converge into one.  The point is novel to the wand and to the cup: rather than an admixture we have a genuine synthesis, as feminine as the emergence in /b/ was masculine.  This synthesis comes not from the urge to express multiple facets but from the experience of having used those facets in conjunction; the overflowing of a full cup.  The outward-extending-into-continuation of the lines implies that we have one wave along a long, waving system.  Indeed, with the type of synthetic awareness indicated in this glyph, a key mode of thinking is to experience oneself as a periodic entity - no more a privileged force of nature than all other entities, who must also be periodic, but recurring as part of a vaster system than those who exclusively contemplate the finite can imagine.  Here is the fluid in XVII, poured from vessel to vessel - where the woman goes, the fluid is poured, and vice versa.  It is XIV to /b/'s VI.  II's receptivity and IX's cane, together.

The last letter, /t/.  This letter is quite simple, being just a slanting curve, but it is distinguished from the rest of the alphabet in that regard - it is the only glyph that consists of one line without any sharp angles.  With the idea of /s/ as a wave among many, this letter accounts for the endless continuum of waves.  Where the primal beginning, /b/, is a differentiation process (even in conception the blastocyst becomes distinguished from the parents' bodies), /t/, the culmination of /s/, is a unified end; it is also the little death from which /b/ springs.  Like Jormungandr, /t/ wraps around the world, but unlike that serpent, it does so by wrapping around itself, for it is the world, seen without distinction; what besides itself is there for it to wrap around?  On dreams the serpent of no-thing, and so on flow our lives.  This is the way Nuit manifests as Had, and Hadit is the hiding of Nu.  Concavity and convexity can only come conjoined.  Nothing or everything are contained in the distinction drawn by this line.  The slope and curve of the line indicate that it is never fixed, except in that it is fixed to change.  It also has a balanced slope - vertical over horizontal change equals its reciprocal, indicating a balance that emerges from self-symmetry (there being no other, /t/ can only be symmetrical with itself).  The snake in XXI, projecting images of its dream onto the rest of the card.  0 just an instant before it starts.  XIII as continuity veiled in change.  XII as the snake and the projection hanging from it.

These are my current notes regarding the individual meanings of the Enochian letters.  They are not meant to persuade anyone, only to show a path.  The next aim in my work with the alphabet it to find a parsimonious way of determining their octave relations to one another.  In the meantime, it's worth looking at a few Enochian words (picked for the fact that they stick in my mind, not because they show the system in an especially positive light):

PIADPH - indicates an enclosure transmitting over all possible paths a gut-wrenching transformation into a double-bound state of affairs (indicated by PH).  The word means "in the depths of my jaws."
AVAVAGO - indicates a branching series of possibilities reaching far down from the macrocosmic order in order to infinitesimally approach the organic.  The word means "thunders."
ZIR - indicates the spinning energy of contained circumnavigation transmitting into the creation of an enclosure.  The word means "I."
AOEUEAE - indicates a vast, sparse matrix of existent entities, ordered such as to connect their own existence and position with the possibilities of another order of entities.  The word means "stars," and the formula is obvious in light of astrology.
UPAAHI - the word has a symmetrical structure, two As in the middle, two enclosure letters outside them, and two vowels on the outside.  UPA indicates the binding of possibility by the grand fractal order and AHI the as merging into apparent oneness of the potential and the actual (i.e. the process by which the potential rebirths itself from a new vantage by actualizing part of itself), and the word as a whole indicates the combination of the grand container of UPA and the Logos of AHI.  The words means "the highest vessels," and the connection to Binah and Chokmah in the tree of life is apparent.

Thank you for your time and attention, and for exploring this alphabet with me in a novel way.  I'd be delighted to hear any ideas my fellow explorers have on the matter.

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