I recognize that I have been a bit neglectful of this topic, a topic which should be of utmost importance in this immaterial, teleological cosmology.  I am looking back at my pervious attempts on this topic.  One can readily determine that my neglect of this topic has been thoroughgoing.  I have alluded to the eschaton in almost every context, but I have been very short on specifics.  Evidently I have been following the advice of Wittgenstein: about that which we are ignorant, it is best to remain silent.  Now I attempt to break that silence.  Caveat emptor

It would be quite natural to suppose that the best ending would be no ending at all.  I don't quite agree.  I do struggle to make the Eschaton warm and fuzzy, but still it is The End.  If this medicine is too strong, you are welcome to seek solace in the philosophies of eternal return.  Incoherence may seem a small price to pay for foregoing the Eschaton, but I disagree.  I am long on coherence. 

Here is what I can offer by way of warm and fuzzy.  The eschaton concerns history.  History is primarily the sojourn of our egos.  The Eschaton concerns the end of Ego, and what remains?  The eternal self is what remains.  But if you and I were to run into the Eternal Self on the sidewalk, we can be sure that we would not recognize Her.  This is what is most frightening about the Eschaton.  It is far from being transparent to our ordinary ego consciousness.  As far as our egos are concerned it might as well be a brick wall.  Well, the Millennium is our ego cushion. 

It is at this point that the Eschaton is inevitably politicized.  Let's face it, there may be more than a bit of communalism in the eternal self.  The idea of Communion is hardly extraneous here.  Neither is the Hierogamos, or hieros gamos (2100 hits).  I notice six previous 'hierogamos' references in these pages, and also in this post of an exchange with Sarfatti.  In its most abstract form the hierogamos can be represented by the seal of Solomon, and here perhaps even as the Big Six.  

The millenarian communalism of the early Christian communities was something of a prenup, and was later politicized by Marx in his Manifesto.  The comic style Rapture scenarios of conservative evangelicals and Hal Lindsey are another take on the hieros gamos, and they are correspondingly more preservative of our precious egos.  

Is there anything that an immaterial BPW can bring to the wedding feast?  I'll stick by the Big Six as an important ingredient.  The Solar & Atomic elements thereof recall the primal mythos of the hieros gamos of Sky & Earth, Geb evidently has his own prostrate version of Atlas.  The latter event gives rise to a Chicken Little (etc.) scenario of a falling of the sky, poor Geb. 

To my way thinking, the falling of the sky might best be taken as representing an information 'implosion.'  The present explosion of information will either create an information overload, or it will be countered by a self-gravitating coherence.  If there is a cosmic coherence, then, in the almost foreseeable future, a website, not totally unlike this one, will trigger an implosion into the coherence of an immaterial BPW.  Thus do we begin our Millennial awakening to our Eternal Self.  It was We who placed the stars in our holographic sky.  This is the idealist inversion of our Copernican/Darwinian cosmology.  

At this point I can only encourage the skeptics to remain skeptical as to who authored the constellations.  Clearly if our eternal self had wanted to win over the skeptics at an early stage, the constellations could have been arranged in a more orderly fashion.  Or she could have arranged for a more gifted raconteur on the net.  One would hardly advance toward the eternal self by not being true to one's skeptical self, if that is one's natural inclination.  Everything in its own time and place.  



A major weakness in the materialist facade is time.  On second thought I would posit three major weaknesses: space, time and mathematics, and that is not counting its biggest problem: the mind.  Space, time and math collectively constitute the decidedly immaterial foundation of scientific materialism.  Each is an intangible relational system.  Each can only be indirectly measured and perceived.  The alleged objective or mind independent existence of each of these entities remains controversial.  Modern physics has further eroded the substance and foundational nature of space and time, but even the classical view was fraught with paradox.  I have alluded to these problems on numerous occasions here, and I should devote a full essay to the topic, but for the purpose of eschatology, time is of paramount importance. 

I wish to argue that the objectivity of a linear time frame is logically insupportable.  What is not controversial is that time and space as we can either know or imagine them had a beginning.  Until quite recently the scientific consensus expected there to also be an end to space and time.  The fact that there might not be such an end is now the primary mystery or paradox confronting conventional cosmology.  In any case, the notion of an absolute space and time became outmoded almost a century ago.  Space and time are constructed internally to conform to whatever is the cosmological model in question.  An idealist metaphysics must also give an account of space and time.  

It is not surprising that modern idealists do not choose to confront science directly concerning cosmic issues.  Thus do phenomenologists, existentialists and postmodernists focus their attention on the microcosm of egocentric perceptions.  To do otherwise would be to incur the opprobrium of the intellectual establishment.  What may make these pages unique in this regard is that I have nothing to lose except my coherence.  Juxtaposing a mental microcosm with a material cosmos cannot result in anything other than the view of life as an absurdity in a meaningless universe.  Every religious tradition has its proprietary answers to the challenge of absurdity.  I also eschew the religious establishments.  I simply follow the dictates of reason all the way to the end of the BPW.  It is often supposed that the distinguishing feature of we humans is our occasional rationality.  That these few pages are apparently unique in their thoroughgoing exploitation of our rationality is the only aspect of this website that ought to boggle the mind. 

If space and time are not primordial aspects of reality, what is their source?  I have recently been focusing on the cycle as the logical source of space and time and everything therein.  I posit a primordial ouroboric cycle as the source of all phenomena, call it Alpha.  The specific character of this 'original' cycle is probably well beyond our present ability to imagine.  Its own existence attests to a process of symmetry breaking of a dialectical and synthetic nature which progressed from the ancient mythos to the modern scientific logos or cosmos.  The final synthesis of this dialectical process is just the anticipated Telos or Omega.  This would be the end of space and time as we historically have come to conceive of them.  Thus would end our linear history.  

A pressing issue is the time scale from Alpha to Omega.  The most basic fact that idealism brings to cosmology is that the cosmic scale cannot transcend the bounds of the mind.  The only way to maintain the scientific measure of the cosmos would be to invoke a cosmic mind quite distinct from created mind.  This logical dichotomy of Creator and Creation is the dualism which deservedly brings the verdict of irrationality upon the prophetic tradition.  To restore reason is to posit just one mind of which our egos are a temporarily fragmented reflection.  That nature appears to break the bounds of this mutual sentience, can only be an appearance.  But by the same token, it is nature which allows God to prolong Creation by postponing her final revelation.  The self-revealing God is the self-concealing God, and nature is just the instrument of the latter. 

To put it quite simply: it is a given that we will eventually see through nature.  The scale of Creation is measured by the opacity or apparent seamlessness of nature.  That nature is seamless to us is largely due to its being projected by us.  The illusory quality of nature can never be empirically demonstrated outside of specific magical episodes.  The Creator comes to light generally, only by the power of empathic reflection.  These pages are devoted to that end. 

If we could look back toward the Alpha we would see time curling up upon itself, not unlike the curling up of the extra dimensions in the higher dimensional models of physics.  We may expect a similar occurrence as we move toward the Omega.  



This Omega business is not going to be easy.  This is the whole tamale when it comes to an idealist cosmology.  To better understand the Alpha and Omega, we need to know how they relate to the present.  And what is the present? 

The image of the present that is the main competitor to the religious views is existentialism.  This is the image of detached egos lost in space, as if we had been thrown into the world.  I can agree with this up to a point.  One point of agreement is the implicit view that once we were attached to something.  We have not always just been floating in space.  The existentialist might contend that this attachment is simply referring to our prenatal condition.  But I would suggest that, even for the existentialist, something more metaphysical is implied.  After all, the theistic and atheistic versions of existentialism remain close cousins.  It is hard to understand one without the other.  

The idea of being 'thrown' into the world implies that our parturition from wherever was haphazard.  Lately, as we know, this view has been challenged by the Anthropic Principle.  We may have been ejected from something, but there had to have been some care in the process, or we surely would not have survived.  It is not as if we were spawned into the abyss.  The more we learn about the world, the more it appears as a receptacle.  And what can we say about the nature of this receptacle?  



From the point of view of teleology and eschatology we must depart from the materialist view that our receptacle is, relative to our own existence, permanent.  The best we could manage with a permanent world would be an arbitrary attachment and then detachment between it and anything non-material.  This is the incoherence of dualism.  

Our material world then must be impermanent, and its existence must be closely correlated with the existence of mind.  In order for there to be a correlation between mind and matter, there must be an 'evolution' from one to the other.  One must be spun-off from the other.  For the immaterialist, matter must be spun-off from the mind.  This is directly contrary, I readily admit, to almost every appearance.  The appearance of a material evolution is the one major obstacle to idealism.  It is formidable, yes, but is it insurmountable?  That depends on our motivation. 

I claim that surmounting materialism is the only way to go.  We have sojourned nearly as far into materialism as is logically and spiritually possible.  Progress, measured in almost any conceivable manner, would grind to a halt or worse, if we do not keep exploring and pushing on the boundaries of our world.  As the material aspect of our existence has taken center stage, the mental and spiritual dimensions of our world have received negligible systematic, rational attention.  The only way forward now is to construct an immaterialist cosmology, as a counterweight to materialism.  There is nothing else to do now intellectually in order to transcend the increasingly rigid strictures of materialism as well as the decreasing marginal returns of the collective effort we have to put into it.  The sense of coming to a material dead-end may still be subliminal for most of us, but it is pervasive.  We have intellectually neglected the non-material dimensions for so long that we hardly know where or how to start.  We will fumble along in the dark for a while longer, but then the light will begin to shine.  

It is time to focus our powers of reason on an immaterial understanding of the world.  We don't have to look further than Darwin to see where our work has been cut out for us.  How may we transcend Darwin?  We simply have to focus on the immaterialist logic behind the materialist facade of Darwinian evolution.  Why, in heavens name, would there be this apparently egregious contradiction to immaterialism?  The simple answer is that I, and perhaps you as well, can think of no better design plan for this world.  Any other plan would be egregiously arbitrary and incoherent.  This is the greatest irony we will likely have to face.  Materialism is indispensable for achieving a cosmic coherence.  

Let me put this another way.  It is not logically or coherently possible to have just a little bit of materialism.  Clearly materialism has its benefits, just a a system of existence.  For the sake of argument, let us grant that in the BPW the costs of materialism can be offset by its benefits.  

And yet another way: if materialism is to be more than a sideshow, it will be virtually all-encompassing, at least from a phenomenological perspective.  Another caveat of the BPW is that it be taken seriously.  If it is going to entail materialism, it will have to do so in a serious fashion.  The material aspect of the BPW must not be perceived as superficial or as an appendix.  There must be a thoroughgoing adaptation between the material and mental aspects of the world.  Certainly the phenomenology of Darwinian evolution fits this bill.  Is there any other means to this end?  We are probably not in a position to be able to conceive of such an alternative, and, lacking such, we must explore the norms, aesthetics and functionality of materializing the mind in an adaptive regimen under the guise of Darwinism.  Anything short of this would have to be considered a feeble attempt at creation and materialization.  

And that brings up another point.  I guess what I am struggling to say in the above paragraphs is that creation and materialization are practically synonymous.  If God is to earn her stripes as a Creator, she will have to take several pages from the book of Darwin.  Exactly how she sets about doing that, remains to be understood, but I have made a few stabs hereabouts at figuring out the 'mechanics' of materialization.  

But why not, then, just crib all of Darwin?  Why be cute about it?  Why not just let the Big Bang rip, and then sit back and enjoy the show?  



The Big Bang scenario implies a dualism which puts God on the spot and correspondingly would leave the creatures in a much more passive role.  Following Darwin  would place the onus directly on the Creator to work out the connections between mind and matter.  Evolution on each planet would have to be directed, and still there is no guarantee that there would be the necessary mutuality between the material and immaterial evolutions.  Putting them together would be arbitrary and after the fact.  This is the incoherence of deism. 

With theistic idealism, creation is maximally participatory.  A direct realism of perception is built into the more distributed and participatory Creation.  Having an immaterial participatory Alpha, greatly facilitates the logistics of the Omega, which then has considerable symmetry with the Alpha.  

The necessary materialization aspect of Creation is then worked out by means of the Big Six, in conjunction with the related cycles.  Our task on this page is to use the built-in symmetry to get a handle on the Omega.  Now we can speak of a dematerialization.  


Of fundamental importance for Creation is for there to be a graceful exit.  For Creation to be meaningful, there must be a purpose, a teleology, a telos.  The telos must be an integral part of the complete process.  Completeness and wholeness are thereby ensured.  An endless Creation is incomplete.   Such incoherence does not recommend itself to the concept of the BPW.  In large measure, the Omega must reciprocate the Alpha.  



Spinning down the world is a large part of what the Millennium is about.  The cyclic processes are closely associated with our materialization of the world.  We have spun up the world into its present atomistic state.  This is a non-analytic notion at this point, but one that needs pursuit.  To complete the cosmic cycle we need to relax on the sub-cycles.  OK, there is the picture of the spinning figure skater.  Our ego consciousness is like the twirling figure with her arms pressed to her sides.  When she extends her arms back out the spinning is reabsorbed into the cosmic cycle or cosmic consciousness.  We may assume there is an energy and information increase in this reciprocal process.  

Herein lies a conundrum. The faster the microcosm spins, the more linear becomes its space-time manifold.  Spin down the microcosm and the space-time frame will curl up.  Does this make sense?  Don't be too analytical about it: use peripheral vision.  

I should be able to give examples, but I suspect that there is a lot of subtlety and even cosmic censorship in play here.  If this process were too obvious we would be jumping all over it and probably throwing it off its track.  We will be jumping on it only when the time is right.  Nascent trends and processes have to be protected.  This site would be no exception.  Camouflage is in fashion as we stealthily and steadily approach the Millennium and Y2C.  Its OK to rock the boat as long as you don't upset the passengers.  Hiding things in plain sight is a proven tactic.  

After considering the cycles, there is the Big Six and its undoing.  We transcend the cycles and their common points of reference.  From the perspective of the Omega we have the Big Three: Sun, atoms and reproduction.  The solar cycle governs all cycles short of the cosmic one.  The life cycle is the microcosm of the A&O cycle.  Atoms provide a common anchor for most of these cycles.  

[At this point I evidently diverge from the topic of Omega, to be continued later.] 

It may be, however, that the Monster Group does play a role in the 'decycling' of the cosmos, and this accounts for some of the subtlety of the process.  We may have a situation reminiscent of the ancient concern with squaring the circle.  The problem now is the 'rationalization' of mathematics, a feat in which the MG is likely to play a significant role.  In the process we revisit the syzygy of e & pi.  It was that coincidence that lured us into cycles and then to the Big Six.  Come to think of it, that coincidence, when properly understood, may turn out satisfy most of our ancient concerns about the squaring of the circle.  The ancient fear of the Apeiron was, I suggest, a premonition of our fall into the incoherence of materialism.  In the Millennium we struggle back to coherence.  The starting point for that struggle may well be something as subtle as the rationalization of e & pi.  Evidently we will need to study the j-function which is related to the 'near integers' and the Monster, which latter must, in turn, play a role in Anthropics.  It is turning out to be a small world out there in Mathland.  We will tame it on the way to the Omega.  

What I may be struggling to say here is that the observer principle (in this context) must be extended from physics to mathematics.  There is a three way reciprocity between physics, math and reason.  This is a theistic version of mathematical constructivism.  This also figures in the rational 'deconstruction' that we may associate with the Omega.  Cosmos coheres from Chaos.  Mathematics is something like the fossil record of this evolution of coherence.  As we reflect upon the mathematical record of the cosmic coherence, we will achieve coherence to the Nth power.  Our inner coherence will subsume the external coherence going into the Omega.  Just don't ask me to draw a picture of it.  



Some references for the rationalization of mathematics:   

And some general math bookmarks for future reference: 


After reviewing the above sites, I would be hard pressed to provide persuasive evidence for the increasing coherence of mathematics.  I am not sure if this is the fault of mathematics or of mathematicians.  There is a natural tendency  for professionals to carve out niches for themselves, and it is evident that mathematicians are no exception.  Good fences make good neighbors, as a poet once said.  The bulk of our Millennial task will be the removal of fences.  We need only find the proper motivation.  

Ferdinand Toennies (1855-1936) distinguished Gemeinschaft (community) from Gesellschaft (purposive, professional association).  Owen Barfield provides a third and future category of human existence: 'final participation,' a must see.  Egoism or existentialism is the incoherent compromise between materialism and idealism.  This is what passes for rationalism in this dark age between the Alpha and the Omega.  Our least stressful means to transcend ego consciousness would have us rely strongly on our intellectual capacity to see the wholeness of the world and our proper function as its co-creators.  The spirit would then follow up and subsume the intellect.  

The intellectual path to a holistic cosmology will have to include mathematics in a significant manner, and after the fashion of Plato and Pythagoras.  The coherence of physics has come about through its mathematicization.  We now need to rationalize mathematics, but in a theocentric rather than egocentric fashion.  The existence of the Monster Group should motivate us to look beyond our puny egos to a cosmic genius.  

I still seek that elusive observer principle (and here and here).  Godel's self-reference program is part of that picture.  His constructions remain controversial.  



I'm looking at Matthew Watkins 'Prime evolution notes.'  He raises the intriguing possibility that the equal spacing of the integers derives from the exact placement of the primes, popular opinion notwithstanding.  There is at least a mutual evolution between primes and integers.  I would like to know how this might relate to the more obviously necessary evolution of the Monster (and here, here and here).  An observer principle must be at work.  


Why this is the case, became one of the key puzzles and led to a number of arguments with a common gist: evolution is present already at the elementary particle level and the primes allowed by the p-adic length scale hypothesis are the fittest ones.



Savant for a Day: this story appears in this weekend's New York Times Magazine.  It describes the possible temporary benefits of transcranial magnetic stimulation, featuring Allan Snyder's work.  If the results hold up under continued scrutiny, it does not bode well either for Darwinism or mechanism.  Being able to squeeze more performance out of the brain with such crude interference just does not fit our material conception of the world.   There will hand waving attempts to make it so, but some of us will still have to chuckle.  



Here is a quote from J. Brian Conrey, The Riemann Hypothesis (3/2003): 

There is a growing body of evidence that there is a conspiracy among L-functions—a conspiracy which is preventing us from solving RH!  The first clue that zeta- and L-functions even know about each other appears perhaps in works of Deuring and Heilbronn in their study of one of the most intriguing problems in all of mathematics: Gauss’s class number problem. [...]  We begin to suspect that the battle for RH will not be won without getting to the bottom of this conspiracy.

L-functions provide a connection between FLT and RH: as described here and here

Goldston and Yildirim's 'proof' concerning the distribution of primes as indexed on Google....  



Status/hiatus report: 

The re-enchantment of mathematics is the most likely trigger for the Millennium, IMO.  I may not have stated it quite as succinctly, but it is a bush that I have been beating around since last year.  I pick math over physics or psychology.  Those would be the next most likely sources of a paradigm shift.  Mathematics, however, remains the soft underbelly of materialism.  

Immaterialism will have to hit materialism where it hurts and where it is vulnerable.  Psychology is peripheral to fortress materialism.  Psychology has never been considered scientific by 'real' scientists.  They do not consider it to be their concern.  Physics is the key to physicalism, obviously.  The cutting edge of reductionistic physics is high energy physics, but it has been running out of money, and energy.  The bastion of reductionism has thus shifted, of necessity, from the experimental atom smashers to the pencil and paper crowd of theoretical or mathematical physicists.  The crunching of the Riemann Hypothesis has become of greater concern to hard core science than has the smashing of quarks, in my political/psychological estimation, at least.  

Let's look at Riemann, or at least at the primes.  If materialism is correct, the prime numbers are random.  So take a look here (and here, here and here (here!) (explanation?), and recall this, etc.) [Ulam Spiral].  To what extent can patterns such as these be deconstructed or reduced?  This may be the biggest skeleton in the materialist closet.  Will such pictures trigger the Millennium?  Certainly not by themselves.  

The space telescope and supercomputers reveal cosmological and numerical patterns, respectively, that strain our modern credulity in reductive explanation.  So does biology.  But it is in the mathematical realm that our intellectual attention can be most readily concentrated and guided.  We would eventually like to know if mathematics is a science or an art. 



I seem to be hearing contradictory statements about the primes.  

There is a general agreement that there is a deep connection between the distribution of the Riemann zeroes and the primes.  It is also agreed that the distribution of the zeroes can be fully explained or modeled by the distribution of the eigenvalues of random hermitian matrices.  Yet we have the phenomenon of the 'jumping champions', Odlyzko, et al, 1997.   The 'explanation' for this jumping phenomenon turns out to be heuristics based on a conjecture.  

Quoting from Odlyzko: 

Conjecture 1.  The jumping champions are 4 and the primorials 2, 6, 30, 210, 2310,.... 

[...]  There seems to be little hope of making any progress towards a proof of Conjecture 1 without assuming at least a quantitative form of the k-tuple conjecture.  However, as we will show, even assuming the strongest form of that conjecture that seems reasonable in view of our knowledge of prime numbers, we are still left with formidable obstacles that prevent us from obtaining a complete proof of Conjecture 1. 

We have certainly come a long way, thanks to Godel et al., from Hilbert's strictures on decidibility at the turn of the last century!  But my issue here is the compatibility of the Odlyzko conjecture and the alleged randomness of the primes.  I have yet to locate a discussion of this question.  

Check out Lehmer's Phenomenon, and then: An improved bound for the de Bruijn-Newman constant, A. M. Odlyzko, Numerical Algorithms, 25 (2000).  Andy is hot on the trail of something, and this is just his hobby:  

The lower bounds support the conjecture of Newman that L >= 0.  This conjecture says that if the Riemann Hypothesis is true, it is barely true, in that even small perturbations to the zeta function lead to counterexamples.  

[7/22 -- here is another reference.] 

Thoughts on the Riemann Hypothesis -- Gregory J. Chaitin, IBM Research (2003). 

Of the authors of the above four books on the RH, the one who takes Gödel most seriously is du Sautoy, who has an entire chapter on Gödel and Turing in his book. In that chapter on p. 181, du Sautoy raises the issue of whether the RH might require new axioms. On p. 182 he quotes Gödel,* who specifically mentions that this might be the case for the RH. And on p. 202 of that chapter du Sautoy points out that if the RH is undecidable this implies that it's true, because if the RH were false it would be easy to confirm that a particular zero of the zeta function is in the wrong place.

[...]  The traditional view held by most mathematicians is that these two assertions, P /= NP and the RH, cannot be taken as new axioms, and cannot require new axioms, we simply must work much harder to prove them. According to the received view, we're not clever enough, we haven't come up with the right approach yet. This is very much the current consensus. However this majority view completely ignores* the incompleteness phenomenon discovered by Gödel, by Turing, and extended by my own work on information-theoretic incompleteness. What if there is no proof?

I am certainly not a randomness freak like Gregory, but I do agree that the public attention being focused on the RH will force mathematicians to depart from their business as usual and deal with the more foundational and philosophical issues raised by the uncanny precariousness of the RH.  It has the character of a syzygy.  What does this mathematical fragility say to us?  Is this another manifestation of an 'invisible hand' at work?  What is the nature of the conspiracy that is present?  Perhaps the new thinking is already out there: we need only locate it.  If not we must anticipate it.  



What is the significance of the RH?  So what if it fails? 

The RH is not just about the randomness of the primes.  Clearly they are not.  The Ulam spiral and the de Bruijn constant testify to structure that is hidden between zeta and the primes.  

The RH has become a cornerstone for a significant portion of the mathematical edifice.  There is no visible architectural substitute.   Number theory would become chaotic for at least awhile if the RH were toppled.  But even if it does not fail, we must contend with the slimness of its validity.  It is as if the realm of numbers were being deliberately maintained at a critical point on the boundary between order and chaos.  



Searching on "prime numbers" & chaos

Quantum-like Chaos in Prime Number Distribution and in Turbulent Fluid Flows -- A.M. Selvam (1999): 

The model concepts enable to show that the continuum real number field contains unique structures, namely prime numbers which are analogous to the dominant eddies in the eddy continuum in turbulent fluid flows.

Finite Temperature Strings -- Mark J. Bowick (1992) 

We hope that the above exposition has given a flavor of the important connection between arithmetic gases and multiplicative number theory. Much remains to be understood for the case of statistical systems that arise from string theories and conformal field theories. 

Continuing the speculation of Matthew Watkins from above: we could add something to the view of Leopold Kronecker that "God made the natural numbers; all else is the work of man."  If God wished to make the integers wholesale, she could start with a quantum chaotic system with the approximate statistics of the zeta zeros and take the Fourier transform to produce the primes and then fill in the composite integers, making sure that the distributions of the zeros would iterate to produce the equally spaced natural numbers.  Just a thought.  Would it not be undignified for God to have to count that first time with her fingers and toes?  This is just a fanciful way of restating the holographic principle that every bit of the world reflects and contains all of the information in its internal, essential relations with the rest.  



A consistent idealism envisions a participatory universe, all aspects of which are essentially and internally related.  The strong anthropic principle already pushes the envelope of what science can tolerate along these lines.  We must go further.  Idealism and immaterialism will necessarily contain strong elements pantheism, panpsychism and vitalism as part of their monistic repertoire.  A materialist science cannot tolerate such elements.  So much for materialism.  

In a monistic and participatory universe there will necessarily be elements of parapsychology, alchemy and astrology.  This is simply how an immaterial world manages to hang together.  These are logically entailed by a holographic, holistic and, yes, a rational view of reality.  

What I have been entertaining on these pages since at least last year is an idealist version of numerical participation.  Critics will wish to label it 'numerology.'  Well, it was not I who first pointed to the 'unreasonable effectiveness of mathematics.'   

There is a possible conflict between this unreasonable effectiveness and the anthropic principle.  Given the supposed inalterability of mathematics and the mathematical basis of physics, particularly w.r.t. exceptional structures such as the monster group, did God ever have a significant choice in Creation?  

To put this point more succinctly, which came first: mind or math?  If the materialists have all become Platonists in their embrace of mathematical physics, where does that leave us idealists?  We have been elbowed out at the banquet table.  

We will have to take a stand somewhere.  The Riemann Hypothesis looms too large now for us not to have an opinion about it.  Mathematicians and materialists both wish to see it proven expeditiously: business as usual.  The looming presence of the RH could prove embarrassing if not otherwise proven.  How so?  

There is apparently a considerable resistance to treating the RH like the Continuum Hypothesis.  In the case of the CH, it's validity was proven to be independent of the other axioms of set theory, and so we were free to take it or leave it.  The clear consensus was to adopt the CH as an additional axiom.  

With the RH, however, the feeling among mathematicians is that it ought to be provable.  It is too closely integrated to the realm of number theory to be considered, or be proven, to be independent of that body of knowledge.  Failure to prove it would be more like the albatross on the neck: not easy to ignore.  It would raise issues of epistemology and ontology.  It must be real, but we can't know it.  We have to take it on faith!  This recasts the Riemann Hypothesis into the mold of the God Hypothesis.  This would not sit well with the materialists, one can imagine. 

Here is another angle.  You may have heard of the Skewes number, weighing in originally at 10^10^10^34.  This is many more zeros than there are atoms in the universe.  Even if you could turn the universe into a roll of leger paper, you would not be able to write it out.  It has since been trimmed down to a very modest 10^371.  This is the number before which the prime count must first exceed its asymptotic form as given by the prime number theorem, and given the RH.  Even at this modest size, we might never have a computer big enough to actually calculate this first crossing point.  There are known to be an infinite number of such crossings further out. 

If the RH fails, there will be that first zeta zero which falls off the critical 1/2 line.  The first 440 billion zeros @ ZetaGrid as of today are right on the money.  And it took several days to calculate the first fifteen zeros back in 1903.  It does not appear that the Bruijn-Newman constant is in jeopardy of being forced positive by these new results, but I would like to see a plot of its more recent values based on the close zeros data.  The record for closeness is well over a year old now.  There ought to be an alarm bell for the next record.  This effort might be compared with SETI@home.  Which is more likely to have a positive result?  

Although SETI presently has garnered much greater interest on the Internet, in the BPW the ZetaGrid ought to be the way to go.  It could be more ambiguous and thus more insidious, but the eschatological significance would ultimately point to the same Omega.  It has taken awhile for me to get back to the page topic, but there it is.  Think about it.    

There would be considerable irony in such a Y2C event.  Jesus was supposed to come back in white robes on a flying saucer, shades of SETI.  With ZetaGrid, Jesus comes back on a Pentium (R) chip, so to speak.  It would be a much more subtle and less disturbing advent.  But how could we link JC to Lehmer's Phenomenon?  Well, that's one way!  

The first non-Riemannian zero would be the most significant number in mathematics, just short of pi, perhaps.  We could make something of it.  And, unlike pi, its provenance would be much more open-ended.  There is not a closed formula to calculate any of the zeros.  In holographic fashion, each zero contains information about every prime number.  There are various computing tricks to obtain approximate information about their locations.  


Here it gets a little tricky.  Find theoretically or computationally the first non-conforming zero and start computing its deviation from the critical line.  This computation will be an open ended process requiring additional theoretical progress.  I would guess that in this extended process additional coincidences such as that of e & pi would show up.  We would gradually be lead to a more organic or vitalistic view of mathematics, a perspective from which we could understand how the existence and physical role of the Monster Group could be compatible with the Anthropic Principle.  In effect, we would see that mathematics in all its Platonic glory is not exempt from the teleology that governs our macro and microcosm.  

This increase in our understanding of the coherence of the world could come about without the intervention of a deviant zero, but let us afford ourselves just this minimal bit of drama or external stimulus.  If there is an eschatological teleology of mind over matter, there will have to be some such minimal numerological component.  To increase the level of the drama, it could probably be arranged for there to be a more conventional, or SETI style, message somehow embedded within the numerical coincidences.  This much to placate Hollywood.  

One might as easily arrange for there to be a message embedded in the positions of the galaxies, say, and meant to be viewed by the Hubble telescope.  Perhaps, but this would have nothing to say about astronomy per se.  It would necessarily be more epiphenomenal than what might be possible in exploiting a mathematical medium.  It would be too much like just exploiting the SETI medium.  God is known to be arcane. 



Numerology is an almost lost art.  It is due for revival.  Ramanujan was the last great numerologist.  He had a direct knowledge of numbers that was unencumbered by layers of analytical education.  Numbers have been disenchanted to become mere labels, no longer having intrinsic meaning.  We turn away from the e & pi coincidence as we might from seeing a ghost.  Mathematical physics has demonstrated that numbers are bound up in the atoms.  How do they leap from the atoms to the mind of a Srinivasa?  Such genius is possible only if the world is a hologram, waiting for us to comprehend the coherence of the micro and macrocosm.  

Riemann's zeta function and the monster group are just two slices of that hologram.  Each number comes alive inside every other number as part of the universal resonance.  That resonance manifests most dramatically as our Omega.  'e' and pi taken separately are just labels.  Taken together they possess a harmony that even Srinivasa could not completely decipher.  We can and will cultivate that genius once we understand its natural necessity.  It will take something like the drama of a 'misplaced' zeta 0 to trigger our deeper curiosity.  That 'misplaced' zero would act as the irritant forcing us to bend our minds toward the reestablishment of the larger harmony.  If this is not sufficient to set us on course to the Omega, then something else will be.  

We understand the significance of 'little' alpha, the fine structure constant, with a value of ~ 1/137.  We know that if it differed by more than one or two percent, carbon based life would have been impossible in this universe.  This is just a piece of Anthropics.  We don't know how it came to have this magical value.  Some suppose that it is merely a random number in each of an infinite set of universes.  It is what it is because we are here to measure it.  That is weak anthropics.  I am a strong anthropicist.  There is just this one BPW in which everything is held together in optimal coherence, this despite the apparent and superficial pre-Millennial incoherence.  

I suggest that Srinivasa II will find a 'formula' for alpha in terms of e & pi, and then will be able to explain it to the rest of us, along with the meaning of the misplaced zeta 0.  Such a 'formula' would summarize the holographic involution of mind and matter.  No small order, this.  



I'm perusing the class number problem.  This relates to the near integral value of the Ramanujan number:  e^(pi*sqrt(163)).  This might shed some light on e^pi ~= 20 + pi, or it might not. 

Uniform Distribution of Heegner Points -- V. Vatsal, 2001. 

HEEGNER ZEROS OF THETA FUNCTIONS -- Jorge Jimenez-Urroz and Tonghai Yang 



Rational points: 9800 hits. 

It is clear that mathematicians are into rationalization in a big way.  The study of rational points on elliptic curves is major.  But I can't tell you what it is about.  I may be dense, however, there is a dearth of background explanation as the mathematicians go busily about their work.  Are they concerned about rational points just because of the intrinsic challenge, or is there an ulterior motive? 

Why do I care about rational points?  It goes back to the natural and venerable impulse of wanting to square the circle.  It is a desire for closure.  That is the point of reason, not to play on words.  The materialist impulse has been taking us precisely in the opposite direction: embracing the Apeiron that was anciently abhorred.  Is it too late to put the Apeiron back in the tube?  Certainly that would be the consensus view if anyone bothered to ask.  Yet there is this residual interest, whatever its provenance. 

For instance, take a look here: Rational Points on Elliptic Curves.  It is as though someone were trying to refute Fermat's last theorem: A^3 + B^3 /= C^3.  But no word of explanation: quite peculiar.  Maybe I should mind my own business and not poke my nose in other folks playpens.  

But then go here: Overview of "Mathematician's Secret Room".  I am being reminded that number theory is about numbers, after all, not about those nasty irrationals (sic).  It was a terrible day for the Platonists and Pythagoreans in 500 B.C. when Hippasus of Metapontum (Hyper-bridge?)

used geometric methods to demonstrate that the hypotenuse of an isosceles triangle with legs of length one (i.e., sqrt 2, sometimes called Pythagoras's constant) cannot be expressed as a ratio of integers. A number of this type is now called an irrational number.  Legend has it that Hippasus made his discovery at sea and was thrown overboard by fanatic Pythagoreans.

And can you blame them?  Ever since, there has been a tension in mathematics between the rationalists and irrationalists. The rationalists remain stranded in number theory while their irrational colleagues help to explore the, no longer rational, heavens.  Leonhard Euler attempted to bridge this chasm back in the 18th century with his invention of analytic number theory.  His goal remains elusive.  

One answer to the above question is simply: the computer or the information explosion.  This is the commercial motive behind the revival of number theory and the renewed, but still in the closet, interest in squaring the circle.  Yes, Pythagoras dream of the harmony of the spheres was reawakened first by digital music.  We often refer to mathematical physicists as Pythagoreans, but the true Pythagoreans would have thrown them overboard right along with Hippasus.  Only Roger Penrose might have been spared from among them. 

All that we rationalists are asking for is an ouroboric closure.  We will even grant you the Apeiron, but it must be the best possible Apeiron, and is must serve the Ouroboros.  

Number theorists continue to strive mightily to contain the inherent wildness of the primes.  Theirs is truly a mission unto the wilderness.  Among other treasures they bring back is the Euler-Mascheroni constant (~0.111^1/4).  Imagine if it turned out rational.  That possibility could well be related to the disposal of the RH.  

New Math Formulas Discovered With Supercomputers: (p.7) 

In April 1993, Enrico Au-Yeung, an undergraduate at the University of Waterloo, brought to the attention of Jonathan Borwein, his professor, the curious fact that:

[a double sum over inverse integers ~= 17*pi^4/360] 

based on a computation to 500,000 terms. Borwein was skeptical of this finding -- if there was such an identity, why hadn't the theory behind it been discovered by mathematicians centuries ago? Borwein tried computing this sum to a higher level of precision in order to demonstrate to the student that this conjecture really did not precisely hold.  Surprisingly, in subsequent computations by Borwein to 30 digits and by myself to over 100 decimal digits, this relation was upheld. Needless to say, it is rather unlikely that a mathematical relation could hold to such high precision and yet not really be true.

Then see Integer-Relations.  

My objection to this approach is that it sacrifices rationality for precision, which is what got us into our analytic fix in the first place.  This approach would miss the near integers, for instance.  Heegner's insight would be lost. 



Integer relations (632 hits): 

Experimental Mathematics and Integer Relations by Jonathan M. Borwein, 2002: 

Mathematicians increasingly use symbolic and numeric computation, visualisation tools, simulation and data mining. This is both problematic and challenging. For example, we mathematicians care more about the reliability of our literature than other sciences. These new developments, however, have led to the role of proof in mathematics now being under siege. 

[...] Many of my favourite examples originate in between mathematical physics and number theory/analysis/knot theory and involve the ubiquitous Zeta Function, of Riemann hypothesis fame.

One thing they do at CECM is attempt to distinguish between algebraic and transcendental numbers.  On a related issue see Is Visualization Struggling under the Myth of Objectivity?  Also see Euler sums in Math World, and Visible Structures in Number theory -- Peter Borwein.  

Let us note again the j-function.  It is responsible for the near integers associated with our e&pi syzygy, and for the structure of the MG.  This is taking us back to moonshine (and here). 

week173 -- John Baez: 

The dimensions of the irreducible representations of the Monster are closely connected to the coefficients of an important function in complex analysis, called the j-function - this connection is known as Monstrous Moonshine.


[...] what interests me most about this stuff is the whole idea of "exceptional structures" - symmetrical structures that don't fit into the neat regular families in classification theorems. The remarkable fact is that many of them are deeply related. As Geoffrey Dixon put it to me, they seem to have a "holographic" quality, meaning that each one contains in encoded form some of the information needed to construct all the rest! It thus seems pointless to hope that one is "the explanation" for the rest, but I would still like some conceptual "explanation" for the whole collection of them - though I have no idea what it should be.



Modular Form -- from MathWorld

Modular forms satisfy rather spectacular and special properties resulting from their surprising array of internal symmetries.

Langlands program

Modular Forms and Elliptic Curves: Taniyama-Shimura 

For a particular elliptic curve, the number of integer solutions in each clock arithmetic forms an L-series for that curve.

A modular form is defined by two axes, x and y, but EACH axis has a real and imaginary part. In effect it is four dimensional (xr, xi, yr, yi) where xr means real part of x, xi means imaginary part of x, and similarly with yr and yi. The four-dimensional space is called hyperbolic space. The interesting thing about modular forms is that they exhibit infinite symmetry under [modular] transformations....

Richard E. Borcherds by C. S. Rajan 

Even before the monster group was proved to exist, hints of its intricate connections with the theory of modular functions began to appear. It was observed by Ogg that in a certain naturally occurring sequence {Sn} of modular curves, Sp has genus 0 (namely it is the Riemann sphere) for a prime p if and only if p divides the order of the monster group. McKay and Thompson found interesting connections between dimensions of vector spaces with irreducible representations of the monster, with the coefficients of the Fourier series expansion of the elliptic modular j-function.

[...]  An example of a vertex operator algebra is given by the Fock space of a string propagating on a torus. The moonshine module is obtained by combining a twisted as well as an untwisted vertex operator module associated to the Leech lattice, and amounts to a theory of a string propagating on an orbifold that is not a torus.

week 95 -- John Baez 

[...]  Well, in dimension 24, there are 24 even unimodular lattices, which were classified by Niemeier. A few of these are obvious, like E8 + E8 + E8 and E8 + D16+, but the coolest one is the "Leech lattice", which is the only one having no vectors of length 2. This is related to a whole WORLD of bizarre and perversely fascinating mathematics, like the "Monster group", the largest sporadic finite simple group - and also to string theory. I said a bit about this stuff in "week66", and I will say more in the future, but for now let me just describe how to get the Leech lattice.

[...]  However, Conway doesn't seem to explain *why* the Weyl vectors have this ascending form. So I'm afraid I really don't understand how all the pieces fit together. All I can say is that for some reason the Weyl vectors have this ascending form, and the fact that the Weyl vector is also light-like makes a lot of magic happen in 26 dimensions. For example, it turns out that in 26 dimensions there are *infinitely many* fundamental roots, unlike in the two lower dimensional cases.

Just to add mystery upon mystery, Conway notes that in higher dimensions there is no vector v for which all the fundamental roots r have r.v equal to some constant. So the pattern above does not continue.

I find this stuff fascinating, but it would drive me nuts to try to work on it. It's as if God had a day off and was seeing how many strange features he could build into mathematics without actually making it inconsistent.

Re- The Monster and the Leech -- John Baez 

In dimension 26 we get infinitely many fundamental roots forming a Dynkin diagram with one node for each point in the Leech lattice, and edges in a pattern that depends only on the distances between these points in the Leech lattice.



Information in the Holographic Universe:  This article from the current issue of Scientific American points to a problem with most scientific speculation.  In short, it is too little, too late, too superficial and too literal.  It contrasts unfavorably with the more meaning based, metaphorical 'holographic' speculation of John Baez excerpted above.  It would be nice if there were a clear path from physics to metaphysics via quantum information, but there is not.  Quantum information is much too atomistic.  Meaning, on the other hand, is fundamentally irreducible.  The path from materialism to immaterialism must pass through coherence and monism.  It must first recognize the irreducible unity of all being.  Short of that realization, scientific speculation is likely to be counterproductive, as is true in this case.  Cosmic intelligence is to be neither quantized nor quantified. 

It is this problem that has motivated me to look to mathematics as the soft underbelly of materialism.  The physical quantum in this context is largely a red herring.  It is a symptom of the underlying, overlying immaterialism, not its source.  At best it is a doorway from physics into the realm of math.  The Baezian 'holographic' unity of math is much closer to the cosmic 'holography' of the BPW than is the quantum of physics.  

What is the meaning of 'exceptional beauty'?  At this point, no one really knows, but we can take a stab at it.  The crux of the matter lies here with the interdependence of the exceptional structures.  Naively we might suppose that mathematical complexity is epiphenomenal, that it only lies on the periphery of that domain.  This is the impression given by chaos theory, where a very few commonalities are overwhelmed by the innumerable profusion of possibilities.  Such is not the case with exceptionality.  Exceptional complexity is truly sparse, and rather than inhabiting the periphery of the subject, it resides at its core, or so we are being lead to believe.  How do we account for this surprising development?  What does it mean, and how might it relate to the BPW hypothesis? 

These exceptional structures are what appear to account for the organicity of mathematics.  These form the vertebrae of the subject, held together by an underlying 'functoriality'. 

Representation theory - its rise and its role in number theory -- Robert Langlands (1989).  This review is a bit more accessible from a physics background. 

Where Stands Functoriality Today? -- Robert Langlands (1997).  This is a mildly pessimistic assessment of the field.  



Endoscopy and beyond -- Robert Langlands (2000).  Robert finds this endeavor more hopeful than functoriality. 

[...] my goal is rather to persuade the younger, more vigorous members of the audience, that the path to a successful treatment of the number theoretical problems to whose solution functoriality promises to contribute may very well lie through the trace formula and that they had best be prepared to master it; it is not to discuss in any serious way the arguments that enter into the complete treatment of the formula. That would be a task not for a single lecture but for a year’s seminar.

Dimension of spaces of automorphic forms -- Robert Langlands (1963).  His early work on the trace formula

"Trace formula" & endoscopy (102 hits). 

"Trace formula" (6100 hits). 

Selberg trace formula and zeta functions -- Matthew Watkins: 

"On a compact (i.e. closed and bounded) two-dimensional surface of negative Gaussian curvature the classical motion [of a point mass] takes place on the geodesics, and it is as chaotic and nonintegrable as possible (being Bernoullian). On this surface there exists also a well-defined quantum dynamics, where the Laplace-Beltrami operator (the invariant Laplacian) acts as the Hamiltonian in the Schrödinger equation. A limiting procedure, exactly parallel to the semiclassical tradition in ordinary quantum mechanics, takes the quantum theory into the classical one when the energy E becomes large, E-1/2 playing the role of Planck's constant... If in addition the curvature is constant, this semiclassical transition is even understood in a certain sense, exemplified by the Selberg trace formula. This formula, which was motivated by Riemann's zeta function, relates in an exact way the spectrum of the quantal motion on compact surfaces of negative curvature to the classical motion. The so-resulting mathematical literature has deep connections with manifold theory, automorphic functions, number theory, etc..."

-- A. Voros and N.L. Balasz, "Chaos on the pseudosphere", Physics Reports 143 no. 3, p. 112.

The resemblance between this formula [Selberg's trace] and the Riemann-Weil explicit formula is such that the N{P} correspond to the prime numbers, and the r(i) on the left-hand side (which are directly linked to the Laplacian spectrum of the surface) correspond to the nontrivial zeros of the Riemann zeta function. Consequently, the resemblance is a major source of support for the spectral interpretation of the Riemann zeta function. Put very simply, the spectral interpretation argues that "the nontrivial zeros of the Riemann zeta function are eigenvalues in some setting".

Selberg zeta function and trace formula for the BTZ black hole -- PETER A. PERRY & FLOYD L. WILLIAMS (2001) 

A Selberg zeta function is attached to the three-dimensional BTZ black hole, and a trace formula is developed for a general class of test functions.  The trace formula differs from those of more standard use in physics in that the black hole has a fundamental domain of infinite hyperbolic volume. Various thermodynamic quantities associated with the black hole are conveniently expressed in terms of the zeta function.

Peter A. Perry

Research Statement: Inverse spectral geometry is the study of the geometric content of eigenvalues of the Laplacian on a compact surface or the scattering resonances of the Laplacian on a non-compact surface. These problems model more realistic problems such as target identification through the scattering resonances of the target.  [...] 

Note that inverse (scattering) problems generally have a holographic basis. 



At this point I am stuck on the math, trying to find the thread that leads to organicity, so lets bring in the big gun, which in our case is the Big Six: AORSAM.   

The connecting theme appears to be the cycle, and recall that it was the e&pi syzygy that triggered the present tangent in an effort to resolve the apparent contradiction between anthropics and the MG.  We were then led to consider the cosmogonic role of the cycle.  

For a cycle to be cosmogonic it must break its essential circular symmetry.  It does so by becoming eccentric, i.e. elliptic.  It is precisely the elliptic functions that are at the core of our 'exceptional beauty', or so we are told.  I can barely make out the thread.  

Before the elliptic functions came the circular functions.  These are the good old sine and cosine functions that dominate much of math and physics, particularly on the engineering side.  The next step in complexity beyond the circular functions are the elliptic ones, and these dominate the rest of math and physics.  Between them, the circle and the ellipse have a firm grip on much of our scientific or material reality.  

The specific connection between elliptical functions and exceptional beauty is to be found in the modular form

Modular forms satisfy rather spectacular and special properties resulting from their surprising array of internal symmetries. Hecke discovered an amazing connection between each modular form and a corresponding Dirichlet L-series. A remarkable connection between rational elliptic curves and modular forms is given by the Taniyama-Shimura conjecture, which states that any rational elliptic curve is a modular form in disguise. This result was the one proved by Andrew Wiles in his celebrated proof of Fermat's last theorem

The circular functions are periodic.  The elliptic functions are doubly periodic.  There can be no triply periodic functions, according to Jakob Jacobi.  In some significant sense, the ellipse defines the limits of complexity.  Let us see, therefore, how direct is the path from the 'simple' ellipse to the monster group.  Need we look any further than the j-function

It turns out that the j-function also is important in the classification theorem for finite simple groups, and that the factors of the orders of the sporadic groups, including the celebrated monster group, are also related.

For the explicit modularity of the j-function see Klein's absolute invariant, J, by which it is defined: 

Every rational function of J is a modular function, and every modular function can be expressed as a rational function of J. The Fourier series of J, modulo a constant multiplicative factor, is called the j-function.    

And recall that the j-function is also probably related to our syzygy via class field theory, the class numbers and the Heegner numbers

The Heegner numbers have a number of fascinating connections with amazing results in prime number theory. In particular, the j-function provides stunning connections between e, pi, and the algebraic integers

The lowly ellipse has certainly managed to keep the mathematical pot boiling.  Is there a Chef stirring this pot?  What sort of seasoning has been added to come up with the recipe for the BPW?  I continue to speculate that the answer may lie in our double syzygy: e^pi - pi = 19.999099979....  Here we have a transcendental  manifesting as an almost rational to the second degree.  In that numerical hiccup must lie a tale.  Do I dare plug this into Integer-Relations?   19.999099979189475767266442984669 ==>>.  I ask it to find the minimal polynomial and it returns two very large coefficients.  I plug it into the Simon Plouffe - P&J Borwein inverse symbolic calculator, and sure enough it recovers the above formula.  Then I hit the browse button....  Here are the names that come up:  Thue-Morse and Lehmer.  In the other direction appears Fibonacci and Bessel.  Arbitrarily switching a digit, i.e. 1997909997918947, yields similar results.  I see nothing yet that grabs my attention.   

What does catch my attention is the browser itself.  I had no idea!  Imagine what Srinivasa could have done with this baby!  I'll wager there's a Srini II out there right now, hot on the grail trail of the organic math.  Me?  I'll just trip through some tulips.  

If I plug in plain pi, the nearest hit is ln (Parking + Madelung).  When I try e, the nearest hits are bunch of sums of different integer sequences.  I wonder what the story is there?  With e^pi, I don't see anything striking, but shouldn't I have gotten another 'Parking' ticket, or is my slide rule busted?  It may be due to a shifted decimal.  This could be addictive, however.  Does it do horoscopes?  

There must be an innocent explanation for all those sequences near to e.  It must be rounding errors of some kind.  Otherwise someone is really missing something.  OK, it seems to be hitting on all the nearly sequential sequences of integers, i.e. e = S (n/n!).  Perfectly innocent!  Well, another Fields Medal gets away.  Somehow I think Srini wouldn't have fallen for that one!  Just checking.  

There ought to be a way to filter through these symbol sequences to pull out significant hits and misses.  Has this not been done?  Is this the best we can come up with?  Surely we are missing something.  I'm still wondering if Renyi ever met Madelung before they bumped into each other here at pi?  Are they trying to stir up trouble, or are they just passing the time?  Is there no cop on this beat? 

One might choose to compare the elliptic generator with the Mandelbrot generator.  Both are superficially simple and yet they both yield a nearly unbounded complexity, but is there any doubt as to which is more essential to math?  The ellipse is in a class of its own.  I don't think we fully understand its rationale.  The area of an ellipse is pi*a*b.  The perimeter is something else.  One of its most elegant approximations is pi(a+b)(1+3h/(10+sqrt(4-3h))), due to Srinivasa.  He calculated it on the toes of his left foot.  Imagine pulling that out of the blue, and this is perhaps the simplest formula to appear is his notebooks.  



This has to do with symmetry.  A circle has too much, a blob too little.  The ellipse rules symmetry, certainly in two dimensions.  The symmetry at higher dimensions is a shadow of what happens down here on the flat.  Those extra dimensions can only buy so much.  Beyond twenty-six, the action seems to fall off.  If there is a higher dimensional version of the Leech, it is unable to cast any longer shadow, certainly not onto anything arithmetical, and arithmetic is the ultimate arbiter when it comes to numbers.  

There is a similar thing going on with the octonions.  These eight dimensional matrices define the limits of matroid complexity and symmetry.  What exactly are the lines between elliptic, octonion and Leech symmetry and complexity is not within my present purview, but who is to doubt that those lines have been drawn?  

Numbers are what we have been talking about these last few months, and it is Riemann who still owns them, as much as anyone else.  The ellipse, octonions and the Leech, if they are to give up their secrets to anyone, it will be to the RH.  Will there ever be another RH?  Perhaps not.  After the RH, math will be puzzle solving.  It will be picking up the pieces.  What will we know then?  I'm suspecting we will know something more Godelian than Godel.  We will begin to know the secret lives of numbers, and how we fit into their picture and they into ours.  

Can we not proceed on this not so minor assumption?  Do we have a choice?  Besides, I'll be on a short break here, and need to tidy up around the ranch.  

Along with the RH will go the quantum gravity enigma.  Both are pushing complexity to what appears to be its logical limits.  Both will have to reckon with the Monster Group, and both will have to confront our lowly syzygy.  That latter item might even be the toughest challenge.  

This intellectual chore of tying up our numero-logical loose ends is a principle prerequisite to the Millennium.  Yes, the Omega does cast its shadow upon the Monster, and not the other way around.  The numero-logical key to the Omega and the Monster is almost within the present the purview of the Symbol Calculator.  We just need to know where to start looking.  The syzygy is trying to tell us where and how.  I just don't have the musical ear for it.  Not here.  Not now.  Will Srini II have to be up to speed on the Omega?  Yes, there will have to be some such eschatological motivation and insight to clear this final hurdle, to put it all into perspective. 

It occurs to me that there must be something about Heegner that I'm not getting.  It is this insight that will help to resolve our problem of the numerical coincidences.  I have to remind myself that the same elliptical j-function that generates the almost integers also generates the Monster.  e and pi cast their shadow on the Monster via j.  e^i*pi is only the circle.  Remove the i and we get the j or the ellipse, more or less.  Someone is trying to tell us something here, but they'll have to speak up. 



It is clear that e and pi are the most overdetermined numbers in mathematics.  The are definitely overworked.  What is their provenance?  What is their progeny?  What is their prodigy?  We'll have to find out. 

One of their progeny is i.  It has been noted that e^i*pi = -1 is the most remarkable formula in mathematics.  It is hard to imagine how sqrt(-1) could have existed before e and pi.  It is this foundling that has done more to tame mathematics than any other numeral.  It figures hugely in Riemann's strategy to tame the primes.  Clearly it keeps e and pi on a short leash.  Take away that iota and we get the Monster.  Their iota-less syzygy at 20 is the shadow of the one with i.  

Mixed up in all of this is the j.  According to John McKay, Charles Hermite (1822-1901) invented the j in his transcendental solution to the quintic.  The pentagon, by the way, yields up the golden ratio as nature's premier numeral.  You gotta love Charles' general solution to the quintic in terms of Jacobi's elliptic exponential or theta functions.  

Thus do we see the elliptic j figuring in the differential cubic and the algebraic quintic.  Nothing beyond the three and the five seem to figure in mathematical complexity.  Note that the thetas are 'quasi-doubly periodic'.  Just what does that mean?  What should the 'quasi' signify in this context?  I see the formula, but it doesn't tell me anything.   

The Jacobi theta functions satisfy an almost bewilderingly large number of identities involving the four functions, their derivatives, multiples of their arguments, and sums of their arguments.

What are all these quasi-symmetries?  From whence do they come; whither do they go?  They have to do with the limitation of complexity.  Complexity is bounded by the j.  Why did Srinivasa have to invent a mock theta?  I notice that his name appears in 249 entries at Math World.  Gauss: 277.  Riemann: 308.  Euler: 547.  Jacobi: 225.  Fermat: 249.  Newton: 225.  (Not counting Lie, Abel/Abelian, etc.)  Who is missing from the >200 club? 

The optimization of mathematical complexity must be related the the observer/anthropic problem.  If there were a bigger Monster, might we not have been eaten alive?  We could not have tamed it.  Do not the primes define and contain all complexity?  A bigger or lesser Monster might have rendered mathematical physics less amenable or less fertile, respectively.  There may be limits to diversity.  We may be pushing that envelope.   There may be moral considerations.  



Modular functions

Here is a excellent exposition of the j-function, emphasizing its uniqueness: 

On the Modular Function and Its Importance for Arithmetic --  Paula Cohen (2000).  She refers to a proof  by Chudnovsky of the algebraic independence of pi and e^pi.  What does this imply about our syzygy?  Might we infer that there are other folks out there beating around this same numerical coincidence?  They are being mildly stealthy about it. 

In the same volume of lecture notes is another interesting paper, Algebraic Dynamics and Transcendental Numbers  -- M. Waldschmidt. 



week125: John speaks of the '12-ness' of elliptic curves. 

Harald Cramer and the distribution of prime numbers -- Andrew Granville (1995). 

Unexpected irregularities in the distribution of prime numbers -- Andrew Granville (1994): primes are not purely probabilistic, e.g. mind the gaps.  Taming the primes is no simple matter.  RH not true?  Are the prime numbers independent of each other?  Their peculiar gappiness indicates otherwise.  

The distribution of the primes seems deeply implicated in the core structure of mathematics.  They are part of its alleged organicity.  This is how they are related.  This relatedness should then be manifested by the failure of the RH.   At present my optimal Y2C scenario would be for a theoretical disproof of the RH to be quickly followed by heightened awareness of math's organicity.  Via mathematical physics and the Anthropic principle, immaterialism and the BPWH would finally become established, along with their eschatological implications.  I can imagine no more subtle or benign way to introduce the Omega.  I am suggesting an incompatibility between a fully self-reflexive, organic math and the RH.  This might not be necessary, but involving the RH in this historically consummate turn-around would be my concession to a minimalist drama.  

A failure of the RH would, at first, strike many mathematicians as a blow against mathematical coherence.  I submit that the studying of the manner of that failure, however, will lead them toward a much deeper sense of the coherence between math and the BPW.  The breakdown of the RH would be part of the vital loophole that allows the abstractions of math to be 'cognizant' of, or commodious to, life and mind. 

The 'unreasonable effectiveness of mathematics' is part of an ontogenetic bootstrap.  It is a bootstrap in which the cosmic mind must participate, and we as well, through that mind.  

It is indisputable that conjecture is playing an increasingly vital role is the advance of mathematics, particularly in number theory. This may be as much a symptom as a cause of the organicity of math.  This reliance on the conjectural approach does render mathematics more unstable and more amenable to changes of philosophy.  It is as if mathematics were approaching a critical point antecedent to a phase change in its structure.  We just don't realize yet how we will be fitting into this larger picture. 



A question that has struck me as significant is the relation between the prime numbers and the Monster Group, or, more succinctly, the connection between the zeta function and the MG.  The shortest path between zeta and the MG is probably the 'moonlit' one, i.e. the modular elliptic j-function.  Is there an elliptic aspect of the zeta?  In 1937 Erich Hecke showed (also see the above) that if the Fourier coefficients of a modular cusp form are plugged into a Dirichlet L-series, then that series can be uniquely factored into an Euler product, i.e. a generalized zeta function.  It was mainly Taniyama who connected the elliptic and modular functions, and his result was used by Wiles to prove Fermat.  The moral of this moonlit excursion is that mathematical complexity has an ultimate source: be it the primes, the Monster, the j, or something beyond all of them.  That source is vital to our 'physical' existence via anthropics.  

The primes are not random, as we have seen above with Ulam, Odlyzko, de Bruijn, the 'conspiracy', etc, but presently I am rereading Conrey.  Much of the zeta problem comes down to finding elliptic curves with many rational points (see p. 352 in Brian's article). 



This is the last gasp before leaving for Ireland this pm.  Our connecting flight to Newark was cancelled yesterday due to thunderstorms, which are expected again today.  

Brian's article is the best I have seen as an overview of the RH.  His main thesis is to demonstrate how convoluted the effort to prove the hypothesis has become.  I am still struggling to absorb his cogent comments.  

Here are some leads from Brian: 

On p. 351, note the connection between the Dirichlet and zeta zeros.  And on p. 353: 

The conjecture of Birch and Swinnerton-Dyer asserts that the multiplicity of the zero of the L-function associated with a given elliptic curve is equal to the rank of the group of rational points on the elliptic curve.

That's about it for now.  Sorry to leave in the middle of this unresolved conspiracy. 



I would like to attempt a better resolution of the above items, but first I must recall some old business.  This is from way back when I first got into 'numerology', from 3-84 to 2-86. 

The only specific items that might presently be salvaged from that excursion involve the metric-English conversion.  According to my notes it was not until 1959 that the U.S. defined the inch as 2.54 centimeters.  I do not know the situation in Britain.  [But see 8/12 on the next page.]  Once again we are confronted with a possible 'conspiracy' when considering the following additional points: 

  1. The Measure of All Things: The Seven-Year Odyssey and Hidden Error that Transformed the World -- Ken Alder, 2002.  
  2. 100/2*1.27 = 39.370... inches/meter.  (3937 = 31*127
  3. 2*31*127^2 = 99999
  4. c = 299792458 meters/sec. 

One has to wonder if the French also had influence from across the channel, and to what end?  And for the record, from the same notes: 

  1. e/phi   =    1.67999005... 
  2. pi/phi^2   = 1.199982...  
  3. pi*phi^1/2 = 1.9990...^2 
  4. gamma^2  =  0.9996.../3

and from before

If mathematicians are not becoming a bit paranoid, perhaps they are not paying attention.  Someone somewhere has been working overtime to square the circle.  Was a member of the French Academy privy to this rationale?  If there were such a rationale it would explain much of the ancient wisdom as captured in the sacred geometry of the megalithic kind.  The megalithic masons were operating on an intuition not unrelated to that of Srinivasa.  

The key to these coincidences is still missing.  If there were a convenient way to search for similar coincidences on the Internet, we might be close to putting 2 and 2 together, but see numerical coincidences (358 hits):  

  1. fine structure constant  alpha = [31 x (PI)^6 ] / (21^5) (2 parts/million) 
  2. Cosmic Numerology -- Ivars Peterson 
  3. Stonehenge- Key to the Ancient World -- Richard Heath 
  4. Re- Path-Integrating over Differentiable Structures... -- John Baez 
  5. Numerical Coincidences --  Bill Debuque, et al.  

Let's pause here.  Bill points out that the e^pi coincidence is related to the fact that (pi + 20)^i ~= 1.  I think this was also pointed out on Math World, but the formulas have become illegible (conspiracy?).  I just checked this page again and the formulas are now more legible. [9/1 -- It is apparently only my browser that is failing.  Finally, I fixed this problem by deleting my Internet cache.]  The latter source now yields: 

The presence of 163 in these coincidences may point to Heegner.  



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