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?
[8/13]
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?
[9/1]
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.
[9/4]
Presently I am reexamining Tony Smith's a site wrt several mathematical issues raised here.
[9/5]
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.
[9/6]
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.
---------------------------------------------
[9/13]
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 'dot.com 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.
[9/15]
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.
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