My diversion into 'numerology' on the previous page was extensive; the most extensive since my two year numerical engagement.  Laying on the floor beside my chair are several hundred pages of handwritten notes from that period.  I have mixed feelings about that effort and how it relates to the present one.  On the one hand there was an almost obsessive desperation to find the 'key', looking for relations between sacred geometry and astronomical data.  Yet, I am certainly not the only one to have been bitten by the Pythagorean bug.  The line between 'monstrous moonshine' (and see here) and outright numerology is a thin one.  Professional mathematicians and physicists are very wary of this occupational hazard.  They are not at all comfortable with the Ramanujan phenomenon.  

Pythagoras has been a constant companion on my sojourn from materialism into immaterialism.  I cannot belittle or forsake that presence.  Implicit in the Pythagorean mysticism is a numerical vitalism or even animism.  What are these numbers that are both alien and familiar at the same time?  From whence cometh the Monster Group, and why does it cast such a long shadow over mathematics and physics?  How could we expect to rationalize the world without rationalizing the Monster.  This rationale must be a vital one.  Therein lies the key to Creation if there be such.  

Numerical vitalism and holism are decidedly contrary to the analytic impulse that still pervades the intellectual arena.  There is strong evidence that we are being led into a mathematical holism despite all analytical resistance.  Every new discovery leads to new and unexpected connections.  The challenge of the Riemann Hypothesis is the case in point.  With every passing day there is more at stake with the RH.  It provides a focus for much speculation.  Any serious immaterialist can hardly avoid participating.  

I am speculating that the resolution of the RH will necessarily involve more than mathematical business as usual.  It will require a much more robust acknowledgement of the organicity of mathematics.  In particular it will involve getting a better handle on some of the numerical coincidences considered previously.  This in turn would entail at the least a greatly expanded version of the Heegner type of phenomenon.  This new vision would incorporate important elements of the ancient numerical intuition, some of which may be gleaned from the annals of archeo-astronomy. 

My most outlandish speculation pertains to the evolution of numbers.  Mathematical structure comes about through the 'self-organization' of the number system.  This might be thought of as an extreme from of mathematical constructivism or intuitionism.  Where it differs from these ontologies is in its explicit appeal to teleology.  The teleology of numbers is contained in the overarching teleology of the BPW.  The convolution of these two teleologies is crucial for the overall coherence.  

The eschatological aspect of this 'numerology' comes with the ultimate devolution of this system as it reverts to its primordial organicity.  In this process the numbers will be reenchanted along with the rest of the world.  All of Creation will participate in the reenchanting resurrection or rapture.  I am inclined, quite ironically, to see mathematics as on the cutting edge of this reenchantment.  It is furthest along in recognizing the organicity of its subject matter.  

We have considered the holistic turn in twentieth century philosophy.  The result of this turn, however, has not been an upsurge in coherentism.  Postmodernism has been, rather, a celebration of pluralism.  Coherentism necessarily implies theism.  Professional philosophers are not paid to turn their collars.  Their theological colleagues, in turn, are not paid to engage in the millenarian messianism that would implied in any fundamental turn to coherence.  There are historical, political and intellectual obstacles to a philosophical embrace of anything like the ancient wisdom.  My speculation is that mathematics is much less prone to such considerations.  To the contrary, Pythagoreanism has always been the Achilles' heel of scientific materialism.  

If there is to be a minimalist 'Y2C' event, mathematics represents its optimal venue, IMO.  This is the rationale of my mathematical excursus.  Where I may be able to contribute to such a turn of events is in the elaboration of this rationale and in the collation of its putative mathematical ingredients.  To be the metaphysical point person in such fashion would be no mean feat.  That this role might have historical ramifications should remain an interesting possibility for all of us who are so inclined. 

Let such considerations not prevent us, however, from continuing to pursue this same goal on other fronts as the occasion may demand. 



Please permit me now, retrospectively at least, to attempt to justify my previous diversion from Omega to numbers.  In the last three pages we have traversed from Creation to Omega to numbers.  This would seem to imply that numbers may provide a connecting  link between the Alpha and Omega.  Recall further my admittedly somewhat arbitrary designation of the Big Six, relative to Creation and Evolution:  A&O/Repro/S&A/MG.  There was the Monster Group dangling, not too gracefully, off the end.  Even a rough perusal of these pages would indicate that the Monster has been bugging me from early on.  On the last two pages I have been attempting, fitfully, to turn the tables on the Monster.  I am attempting to exploit the numerical coincidences in rationalizing it.  I recognize that by doing so in such a cavalier fashion, I am going beyond the 'moonshine' protocol and into an almost blatantly 'numerological' stance.  Well, all is fair in love and eschatology.   

Let's focus on the number connection.  I guess my contention here is that it is love and numbers that constitute the keystone in the A/O arch/ark of Creation.  Love by itself is not quite enough.  It needs structural support in the form of logic and its ramifications into a mathematical, anthropic style physics.  It is numbers, along with love, that help to smooth out the gaps and rough edges of Creation.  Numbers and their organicity are essential in bringing about the phenomenal depth of Creation.  The Darwinian-style, global organicity of metabolism, which I crudely lump into the 'Repro' slot in the Big Six, is the other inducer of depth in Creation.  I have yet to fully appreciate how Darwin and the Monster may be mutually rationalized.  This Creation business is a holographic puzzle: any solution will be holistic.  We can only hope that something like the Big Six will provide an apt nexus for our focus.  

What concerns me here is the Logos of Creation.  Logos includes both meaning and logic.  Meaning is the meat and logic the skeleton of Creation.  The natural languages focus on meaning whilst the artificial languages such as mathematics and computer programs focus on logic.  The problem of Creation is to put the meat on the bones, not to be overly elegant about it.  The observer principle may be crucial here.  

The failure of scientific materialism has been just its attempt to put the meat on the bones.  This is also its attempt to squeeze mind and meaning out of atoms.  As we have seen on these pages, this attempt has been abandoned and postmodern pluralism has risen up in its stead.  So perhaps I have misstated the problem of Creation.  Perhaps it should be seen as putting the bones on the meat, again stating it as crudely as possible.  

In the modern view, numbers and atoms are meaningless, per se.  The meaning must be added on after the fact.  This is just a slight realignment of the Cartesian dualism, with numbers now being relegated to the side of atoms and matter.  Thus have we developed mathematical physics to such a fine art for describing pure matter.  

The ancient holistic wisdom has been effectively defiled in the horoscopes of our daily newspapers.  Astrology, numerology and alchemy are pale shadows of what evidently was once a vital art, and which provided much of the rationale for our early civilizations.  The Enlightenment and the Inquisition may be seen as a kind of shadow boxing with the pale remnants of this wisdom.  The Ramanujan phenomenon provides a very narrow and distorted glimpse of the potential power of this intuition.  

The Omega is not about turning the clock back to the 'good old days'.  It does, however, entail a thorough understanding of from whence we come.  The Omega will be like the Phoenix rising out of those inquisitorial ashes.  Without those ashes there could be no Phoenix, whatever may be the consolation of the martyrs.  

Modern mathematics is a fossilized remnant of our onetime wisdom.  It is not easy to reconstruct the vital animal from the lifeless remnants.  We will have to do some serious cribbing. 

Just as a for instance, consider archeoastronomy.  



A lot can and has been done with the sundial.  Then what?  

Next is the moondial.  There are no moondials, but there are lunar calendars, the Chinese being the best known of these, but so were the Greek and Julian calendars, among others.  (Significantly the Greek word for moon, mene, is the root of both mensuration (metrology) and menstruation.)  Why the moon?  Well, because it is there, of course.  But that is a bit too simple.  Behind the fascination with the moon lie two sets of syzygys: the Meton and Saros cycles: 

Meton: 19 yrs * 365.2425/29.53059 = 234.997 lunar synodic months. 

Besides the synodic month there are also the lunar draconic and anomalistic months, as measured by nodes and perigees, respectively.  The latter two are 27.21222 and 27.55455 days.  This last period is due to the elliptic nature of the moon's orbit.  .


242 * 27.21222/27.55455 = 238.99345988... 

223 * 29.53059/27.55455 = 238.99216... 

223 * 29.53059/27.21222 = 241.998689... 

Should these simple facts not add a celestial dimension to our nascent numerical paranoia?  To tell the truth, I had never seen these numbers together until I calculated them just now.  How could one not be impressed?  What did the ancients know, and when did they know it?  How impressionable were they?  

(And that is not the end of it: 223 * 29.53059/365.2425 = 18.0299980... years.  I have no idea what this is about, or even if it has been previously remarked. 18.03 & Saros (80 hits) yields no such remark.) 

Furthermore there is the coincidence of the apparent diameters of the sun and moon, giving rise to spectacular solar eclipses, the most awesome spectacle afforded by nature.    

The moon is unique in our solar system for its large size relative to its host planet.  This fact is sometimes mentioned in the context of Anthropics and in the speculation about the frequency of life in the universe. The simple fact is that estuaries played an important, if not crucial role in the evolution of life.  Tides play an important, if not crucial role in the estuarine biology.  Go figure.  

My contention is that we are very far from being sufficiently paranoid about the mathematical and celestial coincidences that seem to defy mere probability.  My speculation is that they have a common source: the observer principle, not unlike that of quantum physics with Schrodinger's Cat and the rest.  

Of course, none of this makes sense within the restrictions of materialism.  Materialists shun coincidences as if they were being confronted by Banquo's ghost at the banquet.  The shadow of the Monster looms, however, in an utterly unmistakable fashion.  It casts its shadow upon physics and Anthropics.  Might the Monster also be lurking in the lunar shadow?  

Permit me to summarize the 'observer principle': there can be no unobservable universes.  This is partly a tautology, partly a quantum necessity, partly a relational requirement.  It has a theistic, or at least a panentheistic implication.  We suppose that both Creator and creatures are included in this principle.  A principal role of the creatures is that of proxy cosmic observers.  Thus do we have a 'participatory universe', that being the only possible (kind of) universe.  Indeed, with the BPW hypothesis, it is the only universe.  

Even, then, within the rubric of conventional physics and cosmology, the observer principle cannot be limited to the micro scale.  And now, with mathematics playing an increasing role in physics, we have to consider how the observer principle also applies to that whole field.  Thus does our problem devolve to that of the relation between the observer, the moon and the Monster, to put it most concisely.  None is possible without the others, or so we surmise. 

Our friend the syzygy seems to be in the thick of this conspiracy: always showing up at the scene of the crime, as it were. What are these numerical coincidences trying to tell us?   Quite crudely: someone's thumb is on the scala natura.  The thumbprint is remarkably like our own, considering its frequent decimal or digital nature.  Yes, despite Bohr's protestations, God does not play dice with the universe.  The die, at least, have been loaded and we are her proxies in this cabal.  Les jeux sont fait.  

Something that I am definitely not suggesting is that the above coincidences are any sort of quantum effect.  Let us dismiss the possibility that they could be due to a quantum observational bias, as with the watched pot that is not supposed to be able boil, due to that effect.  As an immaterialist, I am after bigger fish, or at least a bigger pot in this case.  Let us remain focused on the Big Six.  The solar/lunar syzygys say something about the provenance of A&O/Repro/S&A/MG, I suspect.  Recall the purported mutual dependence of heliotrope and helios.  That the moon might act as the mensurational companion of the sun is a hint concerning our immaterialist cosmogony.  The moon provides us heliotropic/anthropic co-creators with some extra needed leverage vis a vis the solar cycle; besides its purely, and not to be underestimated, aesthetic role.  It is our handle on the otherwise overwhelming solar dynamic.  By looking for other such handles on Creation and the Big Six, we might find similar syzygys, in which case things might begin to get interesting around here.  Such handles shed light not only on the Alpha, but, more importantly on the Omega, I believe.  The process of spinning up the world will have some connection to its spinning down.  



This began as an addendum to the to the material from day 8/7 on the pervious page.  I attempt to find an answer to the origin of the remarkable exactness of the ratio of the inch to the centimeter which involves the equally remarkable pair of integers: 31 and 127.  One finds peculiar stories from both sides of the channel:  (My speculation proceeds below.) 

From the Amazon book review

[...]Mechain, remarkably scrupulous even in his doctoring of the data, was driven in part by his conviction that the quest for precision and a universal measure would disclose the ordered world of 18th-century natural philosophy, not the eccentric, misshapen world the numbers suggested. Indeed, Alder has placed Delambre and Mechain squarely in the larger context of the Enlightenment's quest for perfection in nature and its startling discovery of a world "too irregular to serve as its own measure." --  Reed Business Information, Inc

Well, perhaps the jury is still out!

and from the Internet:

It may also be noted that in 1964, an agreement was reached between the U.S. and Britain to define the inch as 2.54 centimeters.  Prior to 1964, the inch was defined in the U.S. on the basis that a meter was exactly 39.37 inches long, which led to the inch being about 2.540005 centimeters long, and in Britain the inch was 2.539997 centimeters in length.

Pray tell, from what hat did the Brits pull their strange approximation?

From another source

Even as late as the middle of the 20th century there were some differences in UK and US measures which were nominally the same. The UK inch measured 2.53998 cm while the US inch was 2.540005 cm. Both were standardised at 2.54 cm in July 1959...

Who has the straight story? 


Later development of the British system continued by defining the units by law in the Magna Carta of 1215, and issuing measurement standards from the then capital Winchester. Standards were renewed in 1496, 1588 and 1758. The last Imperial Standard Yard in bronze was made in 1845. 

Pioneer of Precision  ~  Curated by Julian Holland, 1997  (Below are placards from the museum's exhibit:)

Largely forgotten today Captain Henry Kater (1777-1835) conducted pioneering researches in England in the early 19th century to improve the precision of weights and measures. Associated with this was the development of the reversible pendulum for gravity measurements.

[...]  A seconds pendulum is one where the beat (half period) of the pendulum takes exactly one second. It was thought that the length of a seconds pendulum (from the point of suspension to the centre of mass) would provide a basis for reconstructing the yard measure. Using this pendulum Kater determined the length at London to be 39.13929 inches.

[...]  The report of the Commission led to the Weights and Measures Act of 1824 which introduced Imperial Standards. Subsequently numerous copies of the new standards were required, in which Kater worked closely with several of London’s leading instrument makers. [...]  This bar is stamped with the numbers of inches from 0 to 40, but the only precise basis of measurement is provided by fine lines engraved next to ‘0’ and ‘36’. A hand-written label in the lid of the case states: ‘Value from 0 to 36 inches by means of 3 sets in different years - 35.998803 inches of the Imperial Standard Yard’. This is signed by Kater and dated 10 June 1830. The bar seems to have been constructed in the mid 1820s.

[...]  The primary standards of weight and length in England were preserved at Westminster. When the Houses of Parliament burned down in 1834 - an event dramatically captured in two paintings by Turner - these standards were destroyed and new primary standards had to be prepared. Forty bars were cast in 1845 of which one was selected as the primary of the Imperial Standard Yard. Number 18 was supplied to the New South Wales Government in 1855 as the primary standard of the colony. The yard is measured between fine lines marked on a gold pin in the well at each end of the bar. Kater’s work on standards of length in the 1820s provided a significant basis for reconstructing the primary standard.

If plot there be, thick it is. 

Nautical mile. British, Canadian and American inches -- John H Harland

[...]  That law regarded metric units as the fundamental and internationally-accepted standards for the United States. It was this law that formally defined the inch based on the conversion factor of 39.37 inches = 1 metre as stated in the Act of 1866. This ratio gives an inch approximately equal to 25.400 05 mm. In Britain the National Physical Laboratory made comparisons of the Imperial Standard Yard to the International Metre, which yielded differing values for the inch over the years. The 1922 value of 25.399956 mm per inch was arbitrarily selected for use in calibrating the most precise measuring devices. 

Numerical Constants

-- 39.37 "US survey" inches to the meter :  "US Survey" inch.  (1866, 1893)

This equivalence is now obsolete, except in some records of the US Coast and Geodetic Survey.  The International units defined in 1959 are exactly 2 ppm smaller than their "US Survey" counterparts (the ratio is 999998/1000000).  [i.e. 2.54 cm/inch = 39.370078740... inches/meter]

Did nobody then or now wonder about the origin of this (25.4) coincidence?  Was it purely an accident of history?  Were there no thumbprints on those meter or yard sticks?  Do the metrologists sense no ghost in this closet? 

Here is my present speculation.  All the stuff about inches and barleycorns (1,250 hits) is just so much John Barleycorn (15,300 hits) nonsense: humor for the ignorant.  There was something much more serious going on in the metrology closet, something to do with the ancient wisdom.  Since when did 3 barleycorns amount to 25.399956 mm.?  Did this not have more to do with the fact that 2*31*127^2 = 999998?  If the French Academy was truly interested in a decimal system, well, there it is, along with much else.  If you suppose the ancients could not count that high, then consider the Saros.  

There has been a rash of recent books about the history of metrology, but they generally evince a terrific historical chauvinism or nearsightedness.  It is as if the sun had first risen with the Enlightenment and everything before was darkness.  What happened was that analytic thought overtook holistic thought and then became jealous and arrogant concerning its own power and priority.  The barleycorn stories are not woven out of whole cloth, but, really, has not a single historian actually looked at this number: 25.399956?  How could one pass over it without wondering?  Whatever happened to our natural curiosity?  What untold tale of history lies in those three not so innocent nines?  [8/31 -- Is it the three (five) nines or the three barleycorns that truly explain/define the inch?] 

It is not easy to get good information about pre-Enlightenment metrology.  There is surely an academic aversion to the subject which has been exploited by amateurs not unlike myself.  One must read multiple sources and sort though multiple theories and agendas.  Still, it is very hard to come away from the subject without being impressed by the amount of concern and effort that must have gone into the many aspects of archeo-astronomy and geodesy from megalithic times onward.  If those three nines, and the attendant five nines, do not in some manner reflect and even punctuate that ancient punctiliousness, then very strange accidents do happen.  


To round out our story we head back across the channel. -- 

The Assembly approved the proposed unit on March 26, 1791, and work began on realizing it. To replace the hated “royal foot” until the results of the survey were in, a provisional meter was defined, two of which equalled 6 pied, 1 pouce, 10 22/25 lignes of the toise du Perou.


The toise was the primary unit of length in France prior to the introduction of the metric system.

toise du Grand Chatelet:

established in 1668 and said to have been based on half the width of the inner gate of the entrance to the Louvre.

toise de l'Académie:

A French unit of length introduced in 1766 to replace the toise du Grand Chatelet, about 1.949 meters (about 2.1315 yards). It was often called the toise de Perou, because it was used by the Academy's meridian-measuring expedition to Ecuador (at that time Ecuador was part of the Spanish Empire's presidency of Peru). The toise de Perou prototype was an iron bar made by La Condamine in 1735.

toise de Système Usuel

A decree of February 1812 carried this accommodation much further, and established the système usuel.  While calling for the teaching of the "système legal," that is, the metric system, and its continued use by officials, the decree authorized use of  traditional names but now with values much closer to their traditional ones. For most units, defining such values required the use of common, not decimal, fractions of the metric base units, thus breaking a fundamental principle of the metric system.


On November 28, 1798, the French convened an international meeting of experts from friendly powers and puppet states. One of the meeting's committees consisted of four persons, each of whom independently calculated the length of the meter from the measurements made by Delambre and Méchain (and from certain assumptions about the shape of the earth). Their calculations agreed. The meter was established at 0.144 lignes of the toise de Perou shorter than than the provisional meter.

The story now comes down to the discrepancy of '0.144 lignes.'  But, wait, the story is not over yet. 

Since 1795 the former royal jeweller had been producing bars of platinum 4 mm thick, 25.3 mm wide and about a provisional meter long, with plane parallel ends. The lengths of these bars were compared with the length of the meter as determined by the survey. The one nearest that length (at 0°C) was deposited in the National Archives on June 22, 1799, and has since been known as the Mètre des Archives. The metric system itself was legalized on December 10, 1799.

The Mètre des Archives was, by definition, a meter long, from end to end. Metrologists call such a standard an end measure. End measure standards are not a good idea, because any simple way of measuring their lengths requires touching the ends, which causes wear and shortens the standard. A much better form for a standard of a unit of length is a pair of scratches on a metal bar, because the lines' locations can be determined visually. Such a standard is called a line measure.

International interest in the meter and the French proselytizing spirit led to two international conferences (Commission Internationale du Mètre) in 1870 and 1872 to discuss international standardization of the meter. The attendees favored replacing the Mètre des Archives with a new prototype which would be a line measure and made of a harder, platinum-iridium alloy (10% iridium, to within 0.0001%). They also suggested that the meter be taken as the length of the Mètre des Archives, “in the state in which it is found,” without reference to the quadrant of the earth.

In 1875, twenty countries attended the third conference. Eighteen subscribed to a treaty (the Convention du Mètre), which set up the Bureau International des Poids et Mésures. Production of the meter standard, however, proved very difficult. Besides having an extremely high melting point (2,443°C), iridium had not yet been produced in purities greater than 50%. The bars from the first casting of the alloy, in 1874, were rejected in 1877, and the problem was turned over to the London firm of Johnson, Matthey and Co. They succeeded, and one of the resulting bars was made the provisional standard, even though it was 0.006 mm shorter than the Mètre des Archives. In 1882 France ordered thirty more bars, one of which (No. 6) turned out to be, as nearly as could be ascertained, exactly the length of the Mètre des Archives. This bar is the standard which was declared to be the International Prototype of the Meter by the First General Conference on Weights and Measures (first CGPM) in 1889: “This prototype, at the temperature of melting ice, shall henceforth represent the metric unit of length.” The International Prototype continues to be preserved by the BIPM.

(Oh dear, 100 F in Paris with a blackout last night: bring on the ice brigade!  All the more reason for the French to sneer at W.) 

In 1812 the ligne was defined as 1/432 meter.  Then the discrepancy of 0.144 lignes from above is exactly a decidedly non-decimal 1/3 mm.  (I was going to let that 144 = 12^2 pass, but then I saw 4*108.  Don't get us archeo-metrologists started on 108!)  How, convenient and agreeable of the four 'independent' committee members.  

But let us recall that the speed of light, c = 299792458 meters/sec.  The new Delambre meter was 1/3000th shorter, and that made the speed of light 1/3000th 'faster', or the present 299792 km/sec rather than the 299692 km/sec that it would have been with the longer provisional meter.  Just think if he had shaved off one mm rather than just 1/3 mm.  Then this coincidence would have been impossible for the historians to simply ignore.  Or, couldn't we have arranged for the Earth to be just a tad smaller?  



Yes, one can easily imagine a third or fourth nine having emerged out of this intricate choreography.  The back of our historical apathy would have been broken years ago, and all of us would have been born into a brave new world.  So, yes, bring on the cosmic subtlety, and let us work for our keep.  It is more fun this way, don't you think?  God is not going to force the truth down our throats.  We have to go 'Good Will Hunting.'  (Does anyone else out there have a father-in-law who is such an intrepid Good Will hunter?)

Yes, again, we have to use our own heads and imaginations to put two and three together to come up with five.  

All these nines floating around here, isn't it a bit like a Chinese water torture?  Without our missing key, this is all just recreational, and possibly frustrating as well.  (Is someone playing a game with us?)  What could this putative key possibly look like? 



Provenance of the j-function. 

The j-function appears to reside at the overlap of our concerns with those of many mathematicians.  

John McKay: (1/6/03) 

We hear a lot about Klein's work on the quintic - perhaps because of his being a good self-publicist (see The Icosahedron - available in German, Japanese, and English) ... BUT Hermite is not getting the press he deserves. In Comptes Rendus, 1858, vol. 46, Sur la resolution de l'equation du cinquieme degre, he solves the quintic using the modular equation, Phi_5(j(z), j(5z)=0, just as one today solves the cubic using the relation between cos(t) and cos(3t).

By the way, I am involved in writing up the history of the j-fn from elliptic integrals to present day replicable functions. If anyone has pointers or feels they may have some contributions to make, don't be shy in contacting me! 

We'll see about that....

On Math World one can trace the j-function back through Klein's invariant and the elliptic lambda to the Jacobi theta functions

The Jacobi theta functions are the elliptic analogs of the exponential function...

I believe that these are the same theta functions that are used to express the spherical harmonics of angular momentum in quantum physics, and so are doubly(?) periodic on the sphere (torus?).    

It seems to come back to the mother of syzygys: e^i*pi = -1.  What is the rationale of this?  Sure, we can all see the algebraic identities, but is there nothing more to say?   e & pi were discovered independently and long before this identity was discovered.  Did there have to be such an identity?  Does this question even make sense? 

The issue is the connection between e and the trigonometric functions.  e is the main link between geometry and algebra.  It shows up particularly in the Fourier transform, wherein arbitrary functions are expanded as waveforms.  This brings us to the underlying oscillatory aspect of nature.  Often such oscillation is the result of of interaction between exponential processes.  Which came first, the cycle or the exponential?  In Math World it is pointed out that e is minimally transcendental, and so we might suppose that it is more primitive than pi, corresponding to the just mentioned intuition.  In fact, pi has the highest irrationality measure of the listed constants.  The circle is a sophisticated abstraction of natural cycles.  Pi must be constructed from e.  This is what the syzygys are trying to tell us.  

Then why 163?  This simply tells us that pi is related to e through the spherical harmonics of angular momentum.  That is from where the j-function derives, and the rational point (163) of the j-function connects e & pi.  The circle may seem primitive, but only to the abstracted mind.  Creation is not an abstraction.  It is not a mind game.  Creation is about the Big Six, and the opening to the telos.  The Monster Group is part of the rationalization of e & pi through the j-function.  That function expresses the multiple resonances or spherical harmonics of e & pi.  The Monster comes from e through pi and j.  These intuitions are telling us something about how mathematics became realized in physics, or was it vice versa?  We have to develop a feeling for it.  All this comes in time. 



Presently I am reexamining Tony Smith's a site wrt several mathematical issues raised here. 



I am also reviewing the websites of Matthew Watkins, Matti Pitkanen, and Henry Stapp.  

Here is the present situation.  Over the past few days I have been communicating with two individuals.  The first of these (A) is also my first sustained contact through the website.  He and two of his acquaintances have had experiences that appear to be relevant to a messianic dynamic.  There is now a question of whether either of the other two individuals would care to join our discussion.  I have not requested permission to post any of our conversation.  

My second correspondent (B) also prefers anonymity, an occupational hazard in these parts, it seems.  I have corresponded with him on occasion, over a period of several years.  We came in contact through the Sarfatti list.  Tony and Matti are also (former) Sarfatti listees.  'B', Tony, Matti, Henry and I all corresponded several years ago, but I don't recall the specifics. 

If I wish to pursue the present mathematical inquiry, it would behoove me to make my interests known to these individuals.  Even after reviewing their websites, I have almost no knowledge of their theology, if any.  I continue to look for an entrée.  On the other, hand, if there is a renewed exchange with A & Co., that would tend to distract from the math.  We'll see which way this path forks. 



On not hearing anything yet relative to plan A, I would like to proceed with plan B in the following manner: 


An open letter: b 

This is in the form of an informal open letter to the three above named mathematicians.  For the sake of argument and pedagogy please permit me to speak as if Tony, Matti and Matthew were Pythagoreans.  I don't know this to be the case, but I don't think it will fall too far off the mark. 

In these last two pages it seems that I have been developing an argument against strict Pythagoreanism.  This a line of reasoning and evidence that would favor a looser interpretation of that view that might tentatively be labeled as theistic Pythagoreanism.  

This comes back to the question most famously posed by Einstein: Did God have a choice in creating the world?  The strict Pythagoreans would like to be able to answer:  No, God has to follow the numbers.  

This general issue was addressed just a few days ago in the New York Times: One Cosmic Question, Too Many Answers By DENNIS OVERBYE (9-2-03).  

Any theist, as I am, should be gratified that recent trends in mathematical physics seem to be favoring the Anthropic Principle, and thus leaving the door open to a designing Creator, unless we follow the materialists in positing all possible worlds.  Correspondingly, the Pythagoreans are alleged to be chagrined.  

But, on these last two pages, I have been taking a taking a somewhat different path.  I am taking this occasion of an informal open letter to Tony, Matti and Matthew to perform a reality check concerning this path.  First, has the soon to be described view ever been seriously considered before?  If not, why not?  Secondly, is there not any significant cogency to this path?  

From the perspective of the above article, what I am attempting here might be viewed as a strategy to pull the Pythagorean iron out of the fire, and then enlist their support in a combined effort to defeat the Many Worlds argument being proposed mainly by the scientific materialists.  

It is no secret around here that I favor the Leibnizian logic that ultimately there can be just one world and that it must be the best possible world, hence the title of this site.  

Now here is the twist that I am attempting, and which I now bring to your attention: 

At first blush, it might appear that I am reverting to numerology or a numerical animism, if you will, and, in fact, I am, to a first approximation.  I am in the early stages of exploring a holistic or organic interpretation of numbers and mathematics.  It is only at a higher approximation that pantheism or a theistic Anthropics would reenter the picture. 

So where's the beef?  If you quickly scan these last two pages you will see that I am trying to make something out of various types of numerical coincidence.  Moreover, I am attempting, tentatively, to bring these various coincidence under a single rubric. This is necessary.  Without an overarching explanation, even the best of coincidences rapidly dissolve into just so many curious accidents. 


[9/8 -- open letter continued] 

Yes, obviously there is a resonance of numbers, or, more accurately, of structures of numbers.  These resonances are capable of being sensed by the most sensitive of minds and are part of the explanation for mathematical prodigy.  There is also a self-organizing, organic aspect to this phenomenon of numerical coincidence.  This numerical intelligence is an important part of the cosmic intelligence and accounts for the success of Pythagoreanism.  This self-organizational ability is manifested particularly in a singular structure like the Monster Group, and accounts for the 'unreasonable effectiveness' of mathematics in general and of the Monster in particular. 

The numerical coincidences in mathematics, metrology, astronomy and physics point up the fact that numbers have a mind of their own and actively participate in phenomena, and are not just passive or abstract descriptors, after the fact. 

Human and cosmic intelligence actively participate in the self-organizational, organic quality of numbers.  This accounts, in large measure for the Anthropic Principle, and for the fact that the observer effect must be extended from the quantum physical realm to the mathematical realm.  It does not suffice simply to say that God is a mathematician.  There must be a Godelian type of self-referential capacity that transcends any particular formal system.  This ground of being is what accounts for the possibility of cosmic coherence and meaning, and the fact that we could inhabit the best of possible worlds.  Thus does God come in from the cold of a purely mathematical world.  

This is my first stab at making sense of numerical coincidences.  I doubt that it will be the last.  Through this open letter I am inviting others, with similar concerns to partake of a more concerted effort, if such seems appropriate. 




Here is a follow up on the above letter: 

 Matthew Watkins is the sole respondent, to date.  His response is encouraging and we are exploring the possibility of a continuing dialog.  He has academic responsibilities that limit his time.  I am suggesting that we produce a joint communiqué to post here and send to the four non-responders.  No other publication protocol has been advanced.  Matthew has suggested including others in the conversation. 

At this juncture I have had time to read Richard Vitzthum's v Materialism (1995), and here is his online summary.  It was time well spent.  The most remarkable thing about this book is its existence.  By general acknowledgment, it is the first treatise on materialism in well over a century.  The lack of any such recent treatise had simply not occurred to me.  But now that I have read this latest (last?!) one, I can understand the hiatus.  

As Richard readily acknowledges, the rug has been pulled out from under this philosophy.  To paraphrase the witty dentist: the philosophy is fine, but the matter has got to go.  Ouch!  Such is the legacy of 20th Century physics.  But never-you-mind, even without any 'teeth', Richard is going to keep on whistling the tune. 

The disappearance of matter or substance from the scientific lexicon is not news around here.  But it is instructive to see a true believer trying to cope. 

An innocent bystander might wonder that the continued debate over materialism, on which we have dwelt here at length, is just beating a dead horse.  Yes, and no.  If materialists had any horse sense they would at least roll over and play dead.  I just hasten to point out that having this much sense is too much to expect of the true believer. 

We are really talking here, at the BPW, about a coup de grace.  Yes, I have that much sympathy for their terminal agony, but there is a larger purpose, in which they are intended to be the unwitting accomplices.  Materialism exists now as virtually a museum piece in the menagerie of 'philosophies' that constitute postmodernism, or, at least, that is what the postmodernists would have us believe.  But if you look more closely at PM, you will notice a remarkable lacuna: no cosmology.  It is all just variations of existentialism.  The Metanarrative is eschewed. 

In its cosmology, materialism casts a long shadow over postmodernism.  That cosmology is the 800 pound gorilla in the wings, preventing the resurrection of the metanarrative.  The BPW is nothing if not a metanarrative.  If this story is ever to be told, it will have to upstage that cosmological gorilla.  Postmodernism is, most simply, the label we attach to our despair of ever getting out of that shadow.  The despair is ubiquitous, wherever you scratch the superficial bonhomie.  We could put up with being lost in space, as long as our space ship was humming along.  It seems no longer to be humming, or hadn't you noticed?  

Synchronicity (serendipity?) is surely the most prevalent of all 'paranormal' phenomena.  According to the materialists, is not life itself purely serendipitous?  Yes, on the scale of the universe, life is certainly abnormal.  But according to the materialists: sh*t happens.  The Anthropic serendipity is lost on these good folks: they can just postulate an infinite ensemble of random universes.  And they do! 

Such is not the case with mathematics.  If serendipity permeates mathematics, well, yes, that might be a bit of a shock.  It might even stun an 800 pound gorilla. 

Am I suggesting that one simple fact, e^pi - pi ~= 20, is going to overturn three centuries of scientific cosmology?   Isn't this like the flea floating downstream and demanding that the bridges be raised?  

Yes, mathematics could be that subversive to the entire materialist edifice.  And if it isn't?  We'll just have to give it a little push, wont we? 

Richard Vitzthum is already a bit wary of mathematical physics.  Instead of any material substance, or even any spatial void, all we have now are mathematical symmetries that somehow give rise to force fields, when they are broken (somehow). 

But what is all this coziness between math and physics?  We have come a long way from those atomic billiard balls swerving in the dark.  Recall that much of modern physics might be derived from the (astonishing?) fact that e^i*pi = -1.   Is this serendipity, or what?  What multitude of symmetries lie hidden therein?  

I'll just have to quote from Richard (p.220): 

In other words, the formula [Area = pi*r^2] doesn't exist except as vacuous markings or noises outside an intelligence capable of recognizing it. 

That gorilla is looking a bit woozy.  In attempting to steer clear of Pythagoreanism, Richard has (unwittingly?) smacked right into mathematical intuitionism.  Yes, it can be a minefield out there in philosophy land.  Is not any and all symmetry in the eye of the beholder?  And what is physics based upon, other than symmetry?  Is any phenomenon a real phenomenon if not observed?  

Gosh, isn't the universe trying to tell Richard something?  Is he having trouble hearing his own words?  I can hear him, loud and clear. 

Am I making a mountain out of a molehill?  You bet'chya!  That is just how the Creator operates around here in the BPW.  Whenever you see a wagging dog, always look to the tail.  

If there is a plenitude of symmetries buried in e^i*pi = -1, then how many more than that are manifested in e^pi - pi ~= 20?  That little squiggle covers a transcendental degree of possible symmetries.  How do I know?  Let's just call it my mathematical 'intuitionism'.  

It was Steve Weinberg who once said, 'seen one electron, seen them all.'  What truths are uttered from the mouths of babes!  I wish I could have said it first.  But first, let me say, 'seen one number, seen them all.' 

This latter remark could be taken as a paraphrase of axiomatic (Peano?) arithmetic, but look how easily it spins.  This is also the axiom of any holographic, holistic system, is it not?  Imagine the ship-loads of human creativity that have gone into, and are still going into, the evolving meanings in the pair {0,1} of numbers.  The intellectual history of the human species could easily be contained as a footnote to this simple(?) binary.  The boom and bust of the ' bubble' would be a footnote to a footnote.  This is all that Matthew and I are trying to say, 'seen one number, seen them all.'  

Sorry, but I just couldn't resist this little addendum as the dinner bell rings (and I may be the cook!): 

An electron may be represented by its spin vector or 'spinor', (0,1).  The fact that this spinor transforms into -(0,1) under an e^i*2*pi rotation (fermi statistics) is the 'reason' why you and I won't dissolve into gamma rays in the next microsecond.  But, keep your fingers crossed, sometimes those physicists get their signs wrong!  Just a heads-up. 



e^x ~ sum x^n/n!. 

sin x ~ sum (-1)^n*x^2n/2n!.  And we might note that sin (pi/n) has only two rational points: at n = 2, 6.  Recall that the j-function has rational points only at the Heegner numbers (-1, -2, -3, -7, -11, -19, -43, -67, -163). 

Comparing these expansions yields e^ix = sin x - i*cos x, and e^i*pi = -1.  

I am wondering this morning if their might be some significance in the above identities that the professionals may be overlooking.  We all loose our capacity for astonishment at an early age, which may be to our detriment.  

An important related similarity between the exponential and trigonometric functions is their reflexivity under differentiation.  This implies an essential self-similarity.  This is, of course, what the cycle and circle famously exhibit.  

Life would be nothing without its reflexivity and cycles.  How much of life must be contained in e&pi?  More than a little, considering how e&pi are caught up in the Monster and anthropics.  The organicity of mathematics is necessarily focused on e&pi.  Their mutual syzygys simply reflect this fact.  The organicities of math and life must reflect each other in an essential manner that we don't quite grasp.  The observer principle must be built into math the way it is built into quantum physics.  Quantum physics very much revolves around e&pi&i.  Let us not forget i = sqrt (-1).  I recall that Spencer Brown made much of the peculiar reflexivity of i, not wholly unlike its bigger cousin, I.  Recall that the syzygy of e^pi exhibits an essential decimal signature.  Then recall that the golden ratio, phi, is based on the root of the semi-decimal.  The further decimal syzygys of e, pi and phi should not surprise us. 

Mathematics is nothing if not reflexive, even if it was not until Godel that this was logically demonstrated.  Because of this it must be a resonant structure focused on e, pi & i.  

What I am trying to say is that the Pythagoreans and Anthropists are natural allies in the greater scheme of the BPW.  If we can stick together we will show that the Many Worlds gambit of the latter-day materialists is an otiose sham.  We will tame the monster.  We are only lacking Srinivasa II.  And in so lacking, we must anticipate, and in our anticipation is the reality.  This is the Millennial strategy of Y2C.  

It is the imaginary 'i' that bridges (rationalizes) the gap between analysis/dynamics (e) and geometry (pi).  The culmination of complex analysis is the Riemann Hypothesis.  The RH is related to the quasi-modular (doubly periodic) nature of the sum (n^-s) or prod 1/(1 - p^-s).   How do the two sums, x^n & n^x, relate to each other and the larger scheme? 

One question facing us is how this structure might have evolved, apart from the mundane historical perspective.  On the strictly Pythagorean view, evolution is not an option.  We anthropocists are looking for the logical bootstrap which includes cosmic intelligence.  Spencer Brown's 'Laws of Form' includes an implicit evolutionary scheme.  I might have to replace my lost copy of this book.  That imaginary 'i' could be a key element.  Is it not an essential part of the measurement scheme in quantum physics?  The anti-commutator [x,p] =  i*h.  This is just the basic statement of the uncertainty principle.  Position and momentum are dynamically related.  

That 'i' is essential to to q.m. is frequently noted, but has not been intuitively explained, to my knowledge.  In physics, intuition and math are supposed to be separately compartmented.  The essential role of conjecture in both math and physics ought to argue otherwise.  One might argue that 'complex' numbers are merely a convenient formality, which could be replaced by vector notations.  The formalist would say this about every mathematical entity: there is no intrinsic meaning or substance.  It's all a game played with tokens.  Pythagoreans are not impressed by the platitudes of the formalists.  

Our iota is a primary distinction between classical and quantum physics.  As in complex analysis, it is largely responsible for the opening of quantum physics to the dimensions of symmetry.  The merely classical symmetries pale in comparison.  Riemann surfaces or covers of the complex plane are a significant component of physical symmetry.  One of the biggest advances in mathematics was the realization that the real and imaginary roots of algebraic equations could be treated on an equal basis.  The ensuing expansion in our recognition of algebraic symmetries was the main incentive for the invention of group theory to study the new symmetries.  [Could the Monster Group be far behind?]  Also, it is fair to say that fractals essentially exploit the complex plane.  The many connections between the Riemann Hypothesis and quantum physics also depend on this analytic continuation.  

None of these many aspects of the iota seem to speak directly to the observer principle, beyond what is contained in the uncertainty principle.  It may ultimately come back to the label of 'imaginary' that was fortuitously placed on the iota.  A Pythagorean may well ague that this designation was serendipitous.  When, in 1572, Rafael Bombelli introduced complex numbers (not to mention the minus sign as well!) to the world, mathematics crossed its Rubicon.  That single leap from the real to the 'imaginary' on the part of an entire profession remains historically unprecedented.  One may argue that real numbers are an MIR, mind independent reality, but the sqrt (-1)?  Yet, does not the above quantum anti-commutator place the iota squarely back into the MIR?  Are not atoms an MIR?  Well, not according to the 'wonderful' folks of Copenhagen.  The essential role of the iota in the atom, underscores the 'mystical' Copenhagen Interpretation of the quantum.  Most physicists quickly become jaded on the wonder of the quantum.  Theirs is not to reason why!  

I would argue with the Pythagoreans that the imaginary numbers ought really to convince us that all of mathematics must exist in a Platonic realm.  The Platonists, however, were dualists concerning the corruption of the mundane world.  Not so the true Pythagoreans and myself.  We are monistically inclined toward the singular BPW.  Those same Pythagoreans, as we have seen, were reluctant to embrace the irrational, imaginary and transcendental within 'modern' mathematics.  These latter artifacts imply a personal touch.  Finding them, however, fully embedded in mundane physics should instead provoke us to the realization that great minds think alike.  The cosmic mind and the human mind must be mutually reflective, and this does not apply just to Srinivasa I & II.  All of us are invited to bask in the greater glory.  Furthermore, the BPW does not end with mathematics, fantastic and transcendental though it is.  That is just the beginning of the story.  That is just our aperitif.  

Did God and we merely discover the imaginary iota?  I suggest not.  It was the dynamic resonance of life and creation that spun off e, i & pi.  With the materialization of e, i & pi in the observer based physical resonance of creation, the fate of the Monster was sealed.  The Monster Group is indeed the tail-end of the Big Six.  In this case, it is the five that wag the sixth, and not the other way.   There would be no monster without the vital syzygys or resonances of e, i & pi.  There would be no e, i & pi outside of the cosmic creative intelligence.  We are simply taking mathematical constructivism or intuitionism to its logical cosmic end.  It is there that these many philosophies converge.  Let us not tarry in the hinterland.  The Alpha and Omega beckon. 



Dan struggles.  See Dan struggle. 

Don't we all love the constant juxtaposition of idiocy and sapience in our lives.  Its all a part of the aesthetics.  

One might think that, like Scheherazade, I filibuster the eschaton.  That may be, but it is not my conscious intent. 

What I struggle with is the juxtaposition of stark simplicity and utter complexity, in math as everywhere.  In this holographic world, math is a microcosm.  We might think that with math, we are beginning to see the limits of complexity.  That, at least, is one of my hypotheses.  

Is complexity limited in the BPW?  Here are some reasons.  Complexity is not an end in itself.  In the BPW, everything serves the greatest good which is love.  Love is the one thing of which there might never be enough.  For everything else there is an optimum degree.  Complexity is no exception.  Creation cannot be more complex than what can ultimately be comprehended by the creatures.  Every creature must be able to find its way back to the Creator.  This is what coherence is about.  The salvational economy hinges, of course, upon us sinners.  Salvation is mainly our show.  We set the pace for nature.  Coherence and love are our only guides.  Srinivasa II or Y2C are about our recall of coherence.  

In our sojourn into complexity, we have forgotten about the coherence.  We just need to remind ourselves that coherence is possible, that complexity is necessarily limited by the salvation economy.  We can only sojourn half way into these woods.  After that we are on our way home.  

With e, i & pi we see the juxtaposition of simplicity and complexity.  Like three children, separately they are little angels, but together they can create untold mischief, up to and including the Monster.  Their various syzygys or couplings are the path to the monster, but they are also our clues to its taming, to recasting its complexity into the service of love.    

Unlike for e, there are a plethora of formulas for pi.  This probably has to do with its greater complexity or irrationality measure.  Yes, perhaps, I misspoke.  Pi has been able to stir up a pot-load of trouble just on its own.  e & i are responsible for recruiting pi back into the service of the simple cycle, as in e^i*pi = -1.  It is almost a right of passage into the mathematics club to come up with a new and clever formula for pi.  New discoveries in math frequently lead to more such possibilities.  e is easily the most ubiquitous, utilitarian constant, but it is relatively forthright in its appearances.  Pi tends to rely more on stealth and cleverness in its frequent cameos.  e is definitely blue collar, pi is strictly top-hat.  But put them together and add just an iota, it is a whole new ballgame.  The j-function and its modular cousins are quickly recruited to the cause.   

The Monster Group, by its definition, defines the limits of quantitative or 'classical' mathematical complexity.  With the syzygys we are looking ahead to a qualitative, organic, entangled or quantum-like complexity.  In a similar fashion, the genome defines the limits of quantitative biological complexity.  The proteome looks ahead to a qualitative complexity.   This shift from quantitative to qualitative complexity portends our shift from a materialist to an immaterialist understanding of the world.  It signals our return to coherence.  Coherence is how we finally tame complexity.  In order to resolve the Riemann Hypothesis, I believe that we will have to come to grips with this qualitative side of math. 

What will the Pythagoreans make of this shift from quantitative to qualitative complexity?  Will they still be able to say that God is a mathematician?  We may have to redefine what constitutes mathematics.  In the last century, mathematics paralleled philosophy in its shift from the analytical to the holistic.  As far back as 1931, Godel's incompleteness theorem dashed the hopes for a purely axiomatic or algorithmic program for math, even before the computer came on the scene.  Mathematician could not be synonymous with computer.  But now, with this new understanding of the qualitative nature of complexity, mathematics is being removed from its pedestal or intellectual compartment.  The role of conjecture and intuition will have to expand to be more in line with that of other disciplines.  What this indicates is that mathematical intelligence cannot exist apart from a general or natural intelligence.  For instance, the aesthetic correspondences between music and math will have to be taken more seriously.  

What I am pointing to is that, whatever else cosmic intelligence might be, it cannot be considered to be strictly impersonal.  We will have to take more seriously the alleged correspondences between cosmic and microcosmic intelligence.  To be more specific, the shift to qualitative, coherent complexity is a shift not just to immaterialism, but also toward personalism rather than to the impersonalism of the mystics and materialists.  In other words, the coherence of the BPW is necessarily of, by and for persons.  




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