|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.
-- 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
* toise de l'Académie, also known as the toise de Perou
* toise in the System Usuelle, in France between December 10, 1799 and January 1, 1840, = 2 meters
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.