Symmetry
[...]
What immortal hand or eye
Could frame thy fearful symmetry?-- Wm Blake (late 1700s)
In my initial foray into immaterialism I postulated a chaotic primordial state: everything that could happen, would happen without benefit of time or place. Of late, I have turned more toward a primal symmetry.
The symmetry is totipotent. Actuality is realized in the breaking of the symmetry. The breaking could be spontaneous; however, an observable coherence would more likely be self-sustaining, and thus be the dominant process. That this coherence would include an 'aesthetic' dimension would only enhance its observability and so its self-perpetuation. That perception/conception is proactive in all processes has to be a given for any fully relational system.
And why couldn't we have left well-enough (Nirvana?) alone -- end of story? Only a fool would be so naive as to feel tempted by such a question, so just give me time.
It is, of course, impossible for us to conceive of pre-temporal or multi-temporal 'processes,' but, hey, what's a body to do? And, besides, I just work here.
'Early on,' numbers are entrained in the symmetry breaking. Numbers are a significant product of the process. The prime break might be abstracted as {0,1}, with all else contained in the [0,1] interval. Is this not the A & O? Conceivability implies a prime persona, even manifesting as a 'prime mover.' The splitting persona (my poor head!) arrays itself into various configurations, and the resultant relational networks (in terms of small and not so small group dynamics) are the source of all logically subsequent mathematical physics. Such cosmic numerology is the force underlying our Pythagorean world. And if primal symmetry breaking is a rip-off of the Big Bang, well, our hats off to our physical cosmology colleagues. Cosmogony makes for strange bedfellows. Or was it the other way around?
[9/24]
'Originally', psychology, math and physics are virtually indistinguishable. We are now witness to their separate appearing niches, although our continued probing into their foundations provides ample evidence for their primal unity.
Primal symmetry breaking may be viewed as asymmetric cloning or partial mirroring. I can't really define the 'mirror' beyond a general self-reflexiveness of the prime unity, tantamount to mentation. The symmetry breaking must be self-generated, there being no externality to begin with. We are not talking dialectics here, either. At best, this process might be seen as a reverse dialectic. Does this imply that the Omega would be a forward dialectic? Well, OK, if that would keep our Marxist colleagues pacified.
Clonings can give rise to cyclical network dynamics. There is a reproduction cycle, which may be short on metabolism. Perhaps numbers could serve as proto-atoms in a primal metabolic scheme. Atoms, cells and numbers might be indistinguishable in our primordial alphabet soup. They would provide a common currency. More accurately, they would be the necessary product of a relational, circulational network that requires lowest common denominations. Does this imply an economy? Heaven forbid? Problems of conservation and memory arise, but from physics we know that conservation and symmetry go hand-in-hand. There must be countervailing trends of increasing and decreasing symmetry. The '1' of the primal unity is somehow replaced by the '0' or symmetric void of the space-time background. This is surely a trick: like pulling a hat out of a rabbit. Projective geometry and spin-nets might come into play, not to mention holography. Does anyone else feel like there are a few too many balls in the air at this point? A lot is up for grabs in our brave new world.
While all of this is going on, we must not neglect the A/O split, giving rise to our experience of a linear history. It is the sum over relational paths leading from the A to O. Each person represents a partial trajectory. Together we manage to optimize, with quantum-like constructive and destructive interference, the possibilities of our collective trajectory. Interpersonal dream work could be a significant part of this dynamic.
You may have noticed that while carrying on about Creation, I have had little to say about the Creator. My hunch is that our BPW Creator, while being a maximalist wrt Creation, is a self-minimizer. Besides, we are the resurrection of the primal unified persona. No point in getting in our way or unnecessarily interfering in what is a fairly 'natural' salvation economy. It happens even, or especially, while we sleep.
What I seem to be struggling for now is a sense of the creative logos. Yes, we may wish to know, how did the word become flesh. The new testament speaks of a pre-existent logos, sometimes identified with sophia or the christ. The logos mediates creation. This is an odd theistic admission, one might think, but still commensurate with a trinitarian scheme. Am I then to surmise that our friendly Monster was indeed coexistent with the creator, and later to be identified with Jesus? I'm not real sure we want to go there. That is just what I have been trying to avoid.
Rather, the logos must refer to a 'proto-logic' where word and number could be identified. Notice that with words we are still in the realm of indivisibility. Atoms, or literally particles, are logical remnants of the primordial unity. As such do they carry their essential holographic charge. By such do they participate in the telos. The logos mediates both the alpha and the omega. It would indeed carry a salvific charge and so be identifiable with the charism. How then could we distinguish the second and third members: charism (christ?) and spirit? This may have something to do with the bios, specially linked to #2. Quantum entanglement is just the very most superficial aspect of the teleological or even eschatological charism. The quantum mystics have a point, but they have neither the logos nor the eschaton.
Does any of this help us to pin down the logos? The logos may be the closest we come to an external irritant provoking the primal break: the seed for the pearl that is creation. It is the key to the A&O. It would be the essence of cosmic mystery, partaking of the mystery of the cross. Speaking of crosses, we shift to syzygys and the mysterious couplings of eπ all part of the warp and woof of creation.
There are some in the wisdom tradition who identify the logos or creative spark with the overflow of God's love, leading to creation. In such manner we may avoid externality, but how far can we get with the logic of love?
[9/25]
Words are the prototypical exemplars of the logos. I wish to extend that designation to various other significant entities. In the interest of continuity it would be worthwhile to identify the logos as the source of the Big Six. More generally it is the source of the differentiation of the totality. In that sense it may be identified with Spencer Brown's 'mark of distinction.' Note that I have previously identified the logos with the Zim-zum and the syzygy. The logos is rapidly accumulating baggage. We'll have to sort things out a bit.
Of most immediate concern is how mathematical physics may have emerged out of the operation of the logos. This, of course, is the primary point of contact between the logos and the physical or material world: the Pythagorean imperative. The important point is that the operation of the logos must be reversible. The symmetry breaking can be restored, but only after the logos has taken us to the limit of atomization and analysis. The ultimate reversal probably also invokes the third member of the trinity. I am tempted now to identify the third person with the charism of philo-Sophia. Philosophy was intended to be the flip-side of analysis. That philosophy became identified with analysis in the previous century is one of the profound ironies of history. The dramatic turn by which philosophy regains its true identity would signal the advent noted in John 16:12&ff. It is part of the Aquarium protocol that this first advent be identified with the second, sometimes designated here as Y2C. This would be our own little syzygy: logos meets sophia.
The logos is not a dead letter. It is not mere logic chopping. It is the proactive, vital process of creative distinction. It is the only way in which the totality may circulate, recycle itself through a total reflexivity. We are the pivots of that reflection, not the passive mirrors. Astronomers now deploy active-array mirrors: that's us when you add in quantum-like observer effects.
It is a point of fact that mathematics' primary purchase on the physical is mediated at the quantum level. This is where symmetry is given a free reign. The governing structure here is just the e^i*pi = -1 coincidence, or perhaps the numerical mother or seed of all (physical) a/symmetries. When we carve out e&pi, we are carving out a very significant part of the logos. This is a probe into its operation. This little syzygy, as previously noted, captures (or even generates) much of the mathematical corpus through its various ramifications. It is also implicated as a logical source of the observer effect.
Clearly e&pi is no accident. Its reflexivities are crucial to the whole shebang. If e & pi didn't already exist, God would have had to create them, or did she? Therein lies the secret of the logos, or even of the trinity, for heaven's sake.
Am I being too farfetched here? Yes and no, but I'm not finished. One does not have to be a mystic to see some analogies. Pi stands for the totality or matrix, 'e' for the ramifying logos, and 'i' for the charism of the elusive spirit that ties everything back together. Is this 'ana-logos' purely fortuitous? Or is it the necessary projection of the trinity onto the realm of pure quantity? But it is just our numerological contention that there is no such thing as pure quantity. Quantity and quality are ultimately inseparable. We are trying to see how to realign and reunite them, as a significant part of the new philo-Sophia. This is also along the line of taming the Monster.
Quality conceives its own reflection in quantity. This is just the Pythagorean mystique. This is mathematical intuitionism raised to a new level. Quantity must be born out of quality, and we are beginning to see how. The numerical syzygys are the fossils of this birthing or evolutionary process, pace Darwin.
Let's go back a third time to the apparent jumping-off point for this present excursion: this time for the OM, completing the cycle, as it were. In the primal potency all things exist in a virtual state. The primal syzygy is the circulating resonance, often manifested as the OM. If we project this resonance into the virtual realm of quantity, the static duality of e & pi are set into motion by the essential ambiguity of the iota that now connects them. Which came first: the OM or the e^i*pi? The point, rather, is that they are symbiotic. Reification in this case is a bootstrap. The OM is the quality of motion 'before' the quantity of time. The OM is the seed of the Alpha & Omega 'cycle' whose asymmetry manifests the historical linear time of our Metanarrative. This logos is the logical source of physics. The 'numerological' or Pythagorean aspect of it was primal.
[9/26]
I need to be patient here. If this e&pi business has any merit we will need to extract as much information from it as possible. It could be our best handle on a very obscure dynamics.
The above distinction between quality and quantity is facile. It is just that distinction that I am trying to understand. I am trying to understand or justify Matthew Watkins notion of an evolving number system. An organic, evolving numerical system may actually not be anti-Pythagorean, but it is very un-Platonic. That numbers constitute a mind independent reality (MIR) is deeply ingrained, but it becomes a primary obstacle for monism.
There is not, then, simply a projection from quality onto quantity. It should be more like an extrusion of quantity from quality. This extrusion has something to do with symmetry, and it must go far beyond mere enumeration.
Two candidates for the ur-symmetry are the reflective and the cyclic. The former provides a basis only for enumeration. It is the cyclic symmetry that requires attention. Given a relational network of virtual mutually reflective psyches, there will be many possible circuits.... Sorry, but this still is not getting to the crux of the problem. The observer principle must be included explicitly. Now the observer is only implicit as in providing the (mental) basis of the relational network.
In quantum theory the observer is covered by the projection postulate, or by the collapse of the wave function. Looking back again, note the role of the octonions. Hold on while I take another look.
With reflective symmetry, I was using the mirror as a simile for consciousness, but, in fact, mirrors imply the prior existence of Euclidean space, an assumption which hardly conforms with immaterialism. Closer to our needs and assumptions is projective space. It probably is still too closely related to Euclid, but it is a step in the right direction.
[9/27]
It's time to toss a few more ideas into the pot, stirring vigorously. The strategy seems to be to work both ends, quantity v. quality, against coherence in the middle.
An 'early' state of creation is a growing, neurally networked, psychic pantheon. It needs to be shaped into a lattice and then into Newtonian space, along with bodies and their reproductive cycles, a la A&O.
We need a sequential circuit of psyches, a la the zodiac. This latter is just the metanarrative of the A&O written in the sky. This is the original reproducing cycle, somehow resonating and entraining the rest of the neural net. This is the big Om. In 2-D it would likely start with the circle of six, going to the 12 'pack' in 3-D.
In going from the finite lattice to infinite space we progress through the 17 quasi-exceptional space groups up to the final jump-off point at the sporadic monster. In the process we set up, evolve, the primes and integers by fixing 'e' and pi out of fractional dimensions of space and time(s). Transcendental pi determines translational space. Minimally transcendental 'e' determines directional time. Also, in passing from phi (1.618...) (Fibonacci's rabbit sequence) to 'e' (2.718...) we pass from lattice time to continuous time. (e/phi = 1.67999005... (= 2*3*7/5*5) for what it's worth.)
There is your basic dog's dinner, all I need now is an industrial strength blender. Where are Black&Decker when we need them? Just about a half-mile up Goucher Blvd, if they haven't moved to Bermuda.
OK. phi : e = x : pi -> x = 1.870006... (= 11*17/(2*5)^2). In other words: 3*7*11*17/2*5^4 = 3.1416. Then what? Somebody wants to square the circle via Fibonacci. A strange game here.
9/28
If any meaning may be attached to this further coincidence relating pi to 'e' and phi, it indicates a deeper connection between circles and exponentials. We should have already known this from our 'mother' of syzygys: e^i*pi = -1; a fact which we have yet to understand, but of which there are many reminders. We found this reflected in the 'Ramanujan' constant, e^pi*sqrt163, which pointed to a connection between the largest finite group, MG, and the largest uniquely factorizable algebraic discriminant, 163, via the j-function, based on the elliptic generalizations of the e^i*pi circular function, i.e. the functions of quantum angular momentum. The j-function was discovered in the process of factoring the quintic equation. As the cubic is solved by the trisection of angles, so is the quintic, by a five-section. Phi, of course, is the pentagonal number. Pentagonal crystals are quasi-periodic, reminiscent of various other objects of mathematical interest.
In the deck of numbers, the five-spot is more than a bit on the wild side. The power of the decimal system may be one result. Otherwise we would have been duo-decimalists, following the Solar cultists. This goes back to our metric related syzygy of 2*31*127^2 = 999998, and why the inch is exactly 2*1.27 cm . And note that pi^2 ~= 10, and pi^3 = 31.006..., and while you have your calculator handy, try the windows' one with the 'scientific' view, 2*31*127^2 = 9.999996666663888885...^6. Do we think we understand exactly what is going on with these decimals, or are they just playing around? And how could we forget that e^pi - pi = 19.99909998...? I wondered about that pattern of 9s, didn't you? It turns out, after some experimenting motivated by the previous pattern, that 99.982^1/2 = 9.99909996.... Exploiting this fact gives us a recursive formula for pi = ln (pi + 10 + (100 - .018)^1/2). Starting with, say, pi = 3, this formula rapidly converges on pi ~= 3.1415926527.... Pi = 3.1415926535.... The closest memorable competitor is 113\355 = 3.14159292.... You are invited to comparison shop at Math World. This is the simplest, least arbitrary of the bunch. Furthermore, it seems to beg of a functional meaning, as well as underscoring the decimal essence of pi.
But let us wax intuitive. 'e' and phi alone are mere exponentials. Those of us who attend to Malthus have often wondered how to tame these exponentials. Long ago it was discovered that the wildness of the 'iota' (sqrt -1) would do the trick. Now we see a less imaginary way, using the more subtle wildness of the pentacle. 'e' and phi together are able to tame each other. Phi ~ sqrt 5. The above recursion for pi implicitly, but crucially, entails the root of five. Thus does phi entice the exponential to chase its own tail in recursive, ouroboric fashion, yielding a circle as perfect as there are souls in the cosmos: 1 : 10^10. And thus will the pentacle help us to complete the A&O 'cycle', right on time. Can't you just see the teachers handing out tarot decks in math class? What is this world coming to? Srinivasa might understand.
[9/30]
I ask myself, what is it with numbers that could cause/allow them to be so amenable to both humans and physics. From whence derived their evident organicity?
The transition from 'classical' to 'quantum' numbers was very significant. Therein we go from just real 'Newtonian' numbers to complex/quaternic/octonic matricies with an explosion of potential symmetries, along with all the projective implications associated with measurement. From an 'evolutionary' perspective one might imagine this sequence to have been reversed. These complex, higher dimensional, Hilbert-like spaces, or 'twistor' like networks, may have been the breeding ground for what we naively think of as the God-given integers. The integers, especially the prime ones, were somehow the product of a massive, Big-bang-like, observer coordinated breaking of a high-dimensional primordial symmetry. (see Matthew Watkins' animation of the evolution of the prime counting function.) First phi then 'e' and finally pi and would have taken the role of a quark-triplet in alchemically forming, through their numerical coincidences or resonances, the 'periodic' table of the indivisible prime elements.
Admittedly this is a vague and counter-intuitive notion of numbers: the zeros of the Riemann zeta function forming as the discrete spectrum out of an unstructured state. The Monster Group might have formed 'early in the sequence.' The sporadic and exceptional groups, and modular-type generators, could have been the earliest surviving/replicating condensates out of our proto-logical soup, their general structure and functional context preceding the delineation of their individual elements or products. Srinivasa was somehow able to regress his mind to see this evolution and the scribble down some of its fossil remains. The Anthropic principle or observer imperative ensured that the physics had the proper teleological conformation in its co-evolution.
[10/3]
Somewhere on Peter Borwein's organic math website was a quote to the effect that mathematics is something we never really understand, it's just something we do. The idea of organic math is to restore understanding.
I'm still trying to understand e^i*pi = -1, the purported mother of all coincidences. The underlying formalism consists of infinite sums of the form sum (x^n/n!) and sum (-1^n*x^2n/(2n)!). The above identity is formally obvious in these terms, but does it make sense? Can it make sense? The Greeks turned away from any such infinite constructions as being unnatural, irrational and reflections of Chaos, like the square root of two.
Elementary geometry is a model of rationality. Algebra, however, pushes us toward the irrational with its 'radicals' and unsolvable equations. Even with algebra, though, there is, at least, a formal containment.
If, however, we go back to the simple geometric circle and attempt to solve it as an algebraic construction, we fail. It is true that a circle is represented algebraically as x^2 + y^2 = r^2, but if we attempt to find its circumference, we confront a formula with infinitely many parts. With pi we have passed from the algebraic to the 'transcendental' side of mathematics. 'e' shares the same fate, with a similar formula as indicated above. The only known way to rationalize 'e' and pi is to combine them with the imaginary 'iota' as shown. No other combination will do. We are left to wonder why the two numbers reflecting growth and circumference respectively end up sharing this particular fate, and why, in that one particular combination they then undergird so much of math and physics. Much of the organicity of math and mathematical physics appears now to already be latent in this unique combination. This simple fact, true though it may be, has not elsewhere been remarked. To the professionals, this would be like a fish noticing the water. And such was gravity until Newton was hit on the head. One might wish that the recursion of pi = ln (pi + 10 + (100 - .018)^1/2) could play the role of Newton's apple, but it will take more than just another formula to unlock this mystery. We will have to intuit something more substantial about the connection between quantitative and qualitative organicity.
By way of taking up this challenge, there is another item on Peter's website to be noticed. I call your attention to 'Sum 12' in his paper, 'Strange Series and High Precision Fraud'. It is an infinite series of the form sum (e^(n^2/10^10) and it provides an approximation for pi that is correct to 42 billion (!) digits before it fails. Surely this should blow mine and every other approximation for pi right out of the water. It is a trick, but it is a subtle one, based on the elliptic extension our e^i*pi coincidence in a fashion similar to that for the 'Ramanujan' constant. As such, it cannot tell us anything more about pi than its special role in modular functions, something that we already know. Or do we? Even the 10^10 that appears in Peter's 'fraudulent' formula is entirely arbitrary. These facts strongly emphasize the extremely robust role of pi (along with 'e') in these modular functions which now dominate much of math and physics. The singular robustness of pi may even indicate that it is the pivot and 'e' is the lever that moves and structures the quantitative world as seen from an 'evolutionary' perspective. The unreasonable effectiveness of mathematics is apparently based on just these two numbers. It is this amazing pair that holds the whole shooting match together. From whence did they come?
You know what I am almost reluctantly being forced to verbalize here is the possible sexual character of our dynamic duo, the yin and yang of mathematics. The gender of pi seems straight forward, and while e's gender may be more ambiguous, its reproductive nature is not. Consider this as reflected in the Big Six. Perhaps we are just beginning to see the connection between the MG and the reproductive cycle. Monstrous Moonshine (867 hits) may be just the libidinous lubricant that we need to get Creation rolling. Do 'e' & pi have something to tell us about the birds and the bees, or is it the other way around?
[10/4]
Numbers entered the human psyche via the calendar. The earliest calendars were lunar, although it was the solar days that were being counted. The role of the sun & moon in the formation of our most primitive systems of enumeration is not unlike the role of 'e' & pi in modern math. Elaborate arithmetic was necessary to harness the celestial cycles; numerical coincidences pointed the way. Modern mathematics has developed around the interactive dynamics of 'e' & pi, exploiting their coincidences. This analogy may be a stretch, but we will have to stretch our minds in order to grasp the cosmic coherence.
The gender of sun & moon are virtually universal, even in matriarchal contexts; but can we relate this fact to their numerical cousins? To a degree. Pi, with help from the iota, is the more subtle pivot around which the 'explosive' force of 'e' is entrained. The 'Ramanujan' coincidences of 'e' & pi logically point to the elliptic functions, as the Saros coincidence points to the ecliptic cycles. And, yes, was not the draconic cycle our introduction to the ellipse? As the solar eclipse is the mother of all celestial syzygys, so is e^i*pi wrt the mathematical realm, each with their own intimations of fertility. As above, so below.
Might we not also compare the sequential discovery of the sporadic groups to that of the outer planets? Subtle 'gravitational' effects were the tip-off in each case.
With idealism we seek to reverse the roles of epistemology and ontology. The challenges for idealism in the celestial and numerical realms will be similar. We may suppose that lessons learned in one realm will apply to the other.
[10/5]
What we are lacking in the case of numbers is the analog of the evolutionary coordinating role associated with gravity, in the sky. Yes, we do have logic, but I am looking for something beyond that to explain the supra-logical organicity. I am also looking for the numerical analog of the Anthropic principle. It has been suggested that quantum symmetry provides a coordinating link between numbers, physics and observers. But perhaps we need look no further than the coordinating influence of e & pi. Can we tease a telos out of this odd couple?
The Monster Group is the point of 'crystallization', a Jupiter size 'snowflake', for all of the 'exceptional' symmetries, as mediated through the elliptical modularity produced by e & pi. This model might even suggest a a pre- or supra-symmetric state out of which our structures of logic and symmetry condense. This crystallization would have been mediated by an anthropic telos. Previously I had thought of this primordial state as one of chaos, but this in not in keeping with the Big Bang analog as now understood through the concept of 'inflation'. It is the quantum principle which helps to sort out the symmetry breaking of the early universe into an orderly sequential process mediated by the quantum based energy-temperature relation of the radiation background. But just because we may choose to exploit the mathematical analogies latent in the Big Bang model of cosmogony, this in no way commits us to accepting its ontogeny as veridical for any purpose other than the mere 'saving' of the present celestial appearances -- not, of course, to belittle those appearances!
Perhaps the telos of e & pi is hiding in plain sight. It may well be, even must be, inherent in the concept of the circle. But is not the circle much too innocent to be carrying so much metaphysical baggage? Is it not our geometrical ingénue, if this is not being too gender specific?
Let me attempt to remove some of that sweet innocence. I am reminded of the story of the wealthy prospective patron who demanded proof of an artist's credentials, only to be presented with a freehand drawing of a perfect circle. The very idea of the perfect circle and the straight line seem deeply embedded in our psyches, well beyond the explanatory regime of neural dynamics or cultural artifacts. This pair is also the 'pi & e' of geometry, no? In keeping with our gender awareness, note that sunlight, in contrast to moonshine, is the prime source for the rectilinearity of shadows and optics. Recall that the sun is visible as a circle only in the event of a total eclipse! The natural symbol for the Sun is the cross, in its many variations.
Perhaps the psyche and the circle are not accidental associates, as a dualistic reading of Plato might suggest. It is to the emphatic monism of Pythagoras that we must turn. Yes, I think we can blame Plato's dualism for the estrangement of mathematics from life, and ultimately, then, for the disenchantment of both. Descartes' version of Plato's dualism further established the dichotomy, only to meet its comeuppance in the Pythagorean revelations of the high energy physicists.
Perhaps the circle is not just an abstraction from some Platonic heaven. It is alive and well at the core of our being, nay, of all being. The living perfection of the circle may well be the living symbol of our Best Possible World which, in turn, is the embodiment of the Alpha-Omega cycle. It is only in Creation that the circle can finally be rectified. She may do more than just represent the telos of that Creation.
In some sense I am identifying the Circle and the Matrix. They cannot ultimately be separated. The solar cults were, in part, a confused attempt to steal the circle from the moon.
In algebra we see the attempt to analyze geometry into 'mere' numbers. But a particular one of those numbers, pi, has evidently seen fit to harness algebra to its own higher and very organic purposes. Am I forgetting that poor, dowdy 'e'? Let's just say that 'e' seems to exist mainly to serve pi, in several manners of speaking.
The observer principle is seen to be latent, and all but forgotten, in the circle. In the recursiveness of the circle perfection and normativity play a singular, inseparable role. Nowhere else is symmetry omnipresent. The only perfect circle in nature exists in the atom where it resides in the form of e^i*pi, under the aegis of the quantum. Does this mean that every atom has to be observed? Let's be careful here.
Is the quantum normative in any strong sense of that word? Modality enters explicitly into quantum physics, while it remains only implicit elsewhere. It enters into the projective nature of every quantum operator. The complexified, infinite dimensional quantum Hilbert space is nothing if not projective. The projection operation is simply a measurement, and the conundrum of the quantum is precisely the 'measurement problem'. Therein resides its normativity. Measurement is nothing if not normative. Measurement must be more than a mere process. It is irreducibly a procedure. Every quantum operator may be defined only in terms of such a normative procedure. The action of any operator can be understood only in those terms.
We have ventured into normativity before, with respect to: cycles, quantum, biosemiotics, scientific realism, etc. The entire scientific enterprise is replete with norms. For it to then turn around and present us with a normless nature is being entirely, if unwittingly, disingenuous. A normless nature is, in a very strong sense, a formless nature. Does this mean a snowflake cannot exist unobserved?
For the time being, I will avoid answering that question directly. I will be a bit more devious.
So let us consider uncertainty. It shows up in many forms. Certainly it shows up in the positions of particles. The location of a particle may be ascertained only at the cost of imparting to it an unknown momentum. The concept of position, however, implies an absolute background of continuous space that in no way enters into the interactions. This, however, is only an approximation. Quantum gravity forces space to enter into every physical process, and so physicists search for a pre-geometric, relational concept of space that would be logically compatible with quantum physics.
Allow me then to bring up another issue: indeterminacy in mathematics. Unless one is a committed Platonist, the value of pi is indeterminate. Would this uncertainty not also come to the fore in the same manner that absolute or continuous space has come to be questioned? This does point to a further logical inconsistency in the quantum formalism, with uncalculated consequences. Because the quantum formalism, so far, produces accurate results, we need not press the issue. Our concern, however, is not with accuracy, but with validity. Scientific pragmatism, i.e. materialism, is the issue.
If mathematics were absolute, and isolated from the rest of reality, then the existence of mathematicians and mathematical physics would be problematic. Of course, both of these should be problematic for the materialist and the Platonist, but we modernists have been willing to cut them a lot of slack in that regard.
There is detectable, however, an ill-formed anxiety about such issues. Yes, there is even anxiety about pi. It is said that the main reason for the computations of billions of digits of pi is a pragmatic one: the testing of computer hardware. I would suggest, though, that there is a specificity and compulsion in this effort that speak to a more inchoate metaphysical anxiety. The notion of a vital mathematics that might, somehow, rise up from its deep slumber, is a ghost which could only haunt the deepest recesses of mathematical consciousness.
In an increasingly uncertain world, is not Pi the very last vestige of unassailable certainty? Is there no wonder that we desire to grasp it as firmly as possible? Who am I to look askance at such a desire?
Mathematicians may attempt to sidestep Platonism by disavowing the existence of a perfect circle, and instead say modally of pi that it would be the circumference of this circle were it, counterfactually, to exist. Or, from an empiricist or constructivist perspective, pi could be posited as the (ideal?) limit of a sequence of increasingly accurate constructions and measurements, not to mention calculations. Does this resolve the ontological issue, or does it just shift it? What then would be the actual value of pi? Is it the outcome of the last, most exhaustive calculation, or is it a hypothetical average taken over some 'standard' or normative usage? The first answer is the one that seems to beg no further questions. All such calculations are based on collections of hypotheses which can hardly be placed beyond doubt. And can it be that usage has no ontic significance? Can one simply point to an unused dvd of pi's digits and declare it to be 'pi'. The disk might have been scratched or altered unbeknownst to the subject. Try as we might, we are bound to fall back on, at least, a covert Platonism if we do not appeal to norms of some practical kind. Norms invariably change. What will be the value of pi when our civilization no longer exists? Do we then appeal to a hypothetical normative civilization? Or will our appeal finally be to Plato? After four thousand years of philosophical debate, the only thing we seem to know for sure is that there are no absolute answers. Computers may compute until the end of time, and still not resolve the basic issue.
Or else? Or else there is the BPW. What, pray tell, would be the best possible value of pi? Is this not a normative question? On its face, yes; but in the BPW, norms may ultimately become ontic. That is the point of it. That is the telos.
In the BPW pi is precisely what we most (coherently!) desire it to be. End of story? Not quite. Without fear of immediate contradiction I can simply say, 'See!' Yes, it would be very difficult (impossible?) to imagine a better value than the one it seems to have. The value it seems to have is the one that best holds the world together, teleologically speaking. Where does this leave Plato's perfect circle? I would say that it brings Plato and his forms down to Earth, where all the best possible action is.
Was not Plato just following the dictum to not store his treasure on Earth where the moth and rust do corrupt? Some folks are thankful that God did not follow this dictum when she sent her greatest treasure to us. We have done our darndest to corrupt it, but apparently we failed.
Given this new found power over pi, should we not be afraid that we will corrupt it? I'm not losing sleep over that. The only thing I lose sleep over is trying to understand and explain how pi can possibly manage to be as wonderful as it seems to be. Yes, there is a note of animism in that remark. We can put our faith in pi to do its very best thing. We can even hold it up, as I surely have tried, as a portent of the potency of the BPW in bringing the light of coherence into all Creation. Not even Plato's farthest heaven can escape that light. Pi can reflect that light better than any other number. I challenge you to find another. It is surely the first among equals. Should we begrudge its singular brilliance? Not when we realize that everything can and will shine with that same light when the coherence is complete. It then was merely a shiny pebble along the path.
Finally then, did God have a choice in creating pi? How can it be the best possible and the only possible? The same question could be asked of the BPW. If there is a coherent answer, it will be the same for both, and for all. In the BPW, to be is to relate. In that regard, Pi just seems to have more than its fair share of being. And can there be too much of that? Will there not be more than enough being to go around?
We attempt to resolve only the thorniest issues. If you do not feel encouraged and challenged to surpass my feeble effort, I will have failed.
[10/6]
I seek coherence. Coherence is everywhere; it can be found everywhere. To the extent that it can be ascertained, and mainly in relation to the information available on the Internet, my efforts, as demonstrated on this site, remain unique in their scope and depth. That simple fact must then be incorporated into the picture that I present to you. Generally I choose to interpret it in the context of the prophetic tradition.
I claim, with scant fear of reasoned contradiction, that the coherence of the world and our understanding of it will be a growing factor in our survival and well-being. There are, of course, competing claims. I find none of them persuasive, certainly not in comparison to mine. The most recent comparable attempt at coherence goes back to Hegel in the early 1800's. I am not even aware of a comprehensive contemporary critique of his work. There have been a plethora of dissections of it over the years, all in service to much more limited agendas. But no one until here and now has attempted to surpass his effort; and only in that context, the quest for a greater coherence, can any previous efforts be fairly criticized. But not to worry, Hegel is not a significant part of my present concern. Wittingly and unwittingly I have incorporated some of his ideas into my own, those few which I have found truly accessible. Otherwise, I have not been terribly impressed. Perhaps I have missed something important. If such an item from Hegel, or anyone else for that matter, has been overlooked, then please bring it to my attention, ASAP. In the meantime, I unabashedly rely on Google to scour the Internet for relevant ideas. This is where the contest of ideas is increasingly being played out. If there are important ideas from the past which have not yet been recounted in plural contexts accessible to Google, then they may very well pass into oblivion; having to be reinvented if necessary.
Of late, at least since 2/26, I have paid considerable attention to numbers in general and Pi in particular. Let me attempt a recapitulation.
I am aware that most people have a positive distaste for numbers. I share their antipathy for the degree to which, in this digital age, we are being distracted and annoyed by the ubiquity of numbers, which otherwise seem so alien to our personal interests, unless it happens to be the numbers on our paycheck.
And, yet, even at the height of our annoyance, there has been a spate of books, nay, even Hollywood movies, aimed at the general public concerning the affairs of numbers and those who study them. There is ample evidence in history of our ancient fascination with numbers, a fascination which remains not so far below the surface. One has only to look at the probably horrendous number of person-hours devoted to the deliberate choosing of lottery numbers to realize that they cannot be all that alien to us.
In my case, I have come back to numbers, from my long ago academic foray into mathematical physics, with much hesitation and even trepidation. I left that field, for the second time(!), almost thirty years ago feeling burned-out.
Science is an essentially quantitative pursuit. That is what is supposed to distinguish it from all other human endeavors. Any critique of science will also have to be a critique of its quantitative nature. The typical such critique is done from the standpoint of both innumeracy and the antipathy noted above. The result, of course, is superficiality. That is not to be the case here.
I am going to be much more devious. I intend to turn the (multiplication?) tables on my former mathematical colleagues by accusing them of insufficient regard for their own subject matter. Their disregard has two main sources. Either they are jaded by routine overexposure, or they have come to treat their subject as a mere game to be played with symbols.
I am confident, however, that their disregard is ultimately superficial. In the books I mentioned above, there is ample evidence of a high regard for the mysterious potency of numbers, even if it may be a very dim reflection of the outright reverence for numbers evidenced in the constructions we now associate with archeoastronomy or sacred geometry.
But, no, I am not here to either preach or practice sacred math. I would consider that idolatry. I leave that to the Pythagoreans, a surprising number of whom, I suspect, still roam these precincts.
Rather, I take up numbers as a case study in the coherence of the world. It so happens that I have come to see numbers not as a source of that coherence, but as one manifestation of it. It is even possible that they will provide our most persuasive port of entry into the whole matter of cosmic coherence. This is an important possibility when it comes to addressing a scientifically literate audience as is my intention here.
How do I intend to pursue this possibility? WYSIWYG: long on words and short on formulas. I am neither able nor willing to meet the professionals on their own terms. I intend, rather, to spin a verbal web around their subject, which they may find more difficult to break out of than it might seem at first glance. They are potentially a very important captive audience. I wish to be the captor. My failings in that regard should not put them off their guard. Much greater minds than mine are and will be headed down this path. Consider this a mere whiff of grapeshot. My goal is simply to reawaken their original sense of awe and to then give it a specific spin.
Like everything else with which science comes into contact, numbers have been stripped of all essence. I attempt to restore some of that essence by viewing them in a larger context. The larger context is simply that of being. To exist is to partake of relational being. Numbers ought to be prime examples of this essence, but they have been so stripped down that it will take some effort even to restore a minimal degree of their true identity.
Quite simply, numbers have been reduced to mere quantities. They have lost their individual identity. To make amends I take my cue from some recently popular books on numbers, and pick up from where they tend to leave off.
Clearly Pi is a number already standing out from the rest. What are we to make of this? What do its recent publicists make of it? Not nearly enough, I claim.
Typically they point out the historical fascination with Pi, and the fact that we now spend thousands of computer hours calculating billions of its digits. I don't see any real attempt at this level to explain the interest. One has to scrounge through the Internet to come up with even a few hints as to any underlying continued fascination. I attempt to bring those together here in a coherent fashion, and use them as a point of departure for some informed speculation.
Pi is anyone's exhibit A in proving that not all quantities are equal, in the most qualitative sense of that term. If this be the case, then we have at our disposal a tool of great power to undermine what has become the intellectual fortress of scientific materialism. Indeed, Pi then becomes the perfect Trojan Horse in our anti-materialist armament and assorted bag of tricks. Pi has been sitting quietly at the foundations of that fortress from its very inception. We have only to give it the proper tweak, and the fortress is in extreme jeopardy. But keep in mind, we are in no way inclined to demolish science. We wish only to plant therein the flag of a higher truth, and call it our own.
We are concerned with the truth of all being. If Pi is real, then it must partake of the universal, relational being. If there even exists such a thing as mere quantity, and I would argue to the contrary, then, if any number can, Pi rises above that paltry level of existence. If Pi can do that, then so to can anything and everything we can put our minds to, and that includes especially our own selves.
Yes, Pi has been ground down by that relentless leveler and deconstructor that parades as scientific materialism. And much more tragically, so have you and I. Are we not just meat machines powered with genetic drives? In coming to understand the true identity and power of Pi, we can begin to regain our own true identities, all the while putting the lie to materialism.
The case that Pi is not merely a quantity is excruciatingly simple. I have already made it and I will reiterate. The much more significant question is why have we not already made a 'federal case' of this fact. Why is it not being shouted from the roof tops? Frankly, I'm perplexed. On this simple fact may well rest the fate of humanity. There must be a lot of people out there who should know better, but who, for various reasons, are looking for truth in all the wrong places. I just hope that Google has a better nose for truth than several busloads of Professors. It may just be professional courtesy not to rock the boat and not to piss in each other's soup. In that case, please forgive my French.
Here's the scoop on Pi. No. Stop. Let me quote from one of my favorite professors, in one of his more candid moods: Ramanujan, modular equations, and approximations to pi (1996) (sec 2: The state of our current ignorance):
In part we perhaps settle for computing digits of pi because there is little else we can currently do.
This from the foremost calculator of those digits! Nowhere have I seen more clearly distinguished the quality vs. the quantity of any number. Emphatically the elusive identity of Pi is seen to transcend any mere quantity, to no matter how many billions of digits we may choose to know its quantity. But is my contention not then trivial? The mere fact that a given number can be calculated indicates that we must possess a generating algorithm that represents that number, quite independently of any digits that it may generate.
OK. Then is any number completely identified by the algorithm which generates it? Yes, but be careful. Identity and identification are not the same thing. I am not my driver's license, now am I?
An algorithm is a mechanism. As such it is necessarily normative. A machine is not a machine unless there is someway to vouch for its proper functioning. Propriety is necessarily teleological, and that is an issue that can be addressed only by an authoritative agent that transcends mechanism. It may be that our goal orientation is illusory, but then so is everything else, and that is then the end of scientific realism, as well as scientific materialism.
Pi is the circumference of an ideal circle, relative to its diameter, as one among many other roles or functions with which it may be identified. An understanding of its functions allows us to agree on various methods for calculating it. We are not terribly surprised when the resulting calculations of the variously derived algorithms are in agreement.
[10/7]
For any given mathematical constant, there exist an indefinite number of ways to calculate it. Each correct algorithm does identify the number, but even all of them taken together cannot exhaust its identity. Thus does the ratio of the circumference to diameter of a circle specify a quantity. But the specification is not bi-directional. The quantity does not specify the entity. No sequence of digits can identify anything besides itself.
Mathematicians use two kinds of equal sign. One stands for equation or equality, usually meaning quantitative identity, as in 1/2 = 2/4. The second sign is logical identification or simply definition, usually printed as three bars, but let me use # to represent this sign. One may not then say that 1/2 # 2/4, because only one of the fractions has been reduced. Thus one could write that Pi = algorithm(k), but it would be incorrect to write that Pi # algorithm(k).
In this case, only rational numbers may be quantitatively defined: R # n/m, where the fraction has been properly reduced. Sqrt (2), however, cannot be identified with anything other than itself.
But is this not a trivialization of my metaphysical or meta-quantitative claim? How might this meta-quantitative distinction be carried over into science?
In science, the distinction I am making is tantamount to scientific realism. That is realism about its theoretical entities. This realism is taken in contrast to empiricism or phenomenalism. Unless they are formalists or constructivists, mathematicians are realists about Pi, as physicists are about atoms or electrons. When confronted about the nature of their realism, scientists take pains to downplay its obvious metaphysical implications. And we cut them that philosophical slack.
Mathematicians are generally much less bashful about their realism. Constructivism seems barely to get off the ground when confronted with the phenomenon that is Pi. Formalism gets hardly any further, and it then has to contend with Godel's overturning of Hilbert's formalistic thesis.
I have to be careful here, myself. I often speak of myself as a relationalist. Radical relationalism posits that only the totality is real. Any attempted subdivision of the totality generates only illusion. I have to admit a degree of sympathy with that view, at least as a last resort against Platonism. Besides, I am an internalist regarding relations, which is the only coherent possibility for the radical version. Pi is real in as much as God resides therein; in as much as God chooses to be manifested in that fashion. This may sound extreme, but there is no other way to make sense of it. Pi may only be understood as a particular rational or teleological projection of cosmic intelligence.
One conclusion to be drawn from the foregoing is the advisability of allowing Pi to stand in for the Monster Group in the Big Six, which then becomes AORSAP. Among other things, this tends to place more emphasis on the metaphysical importance of cyclicity, which on the cosmic scale is taken simply as the closure of the A&O. Hereby are we afforded considerable simplification.
A fuller appreciation of the supra-quantitative nature of Pi allows me to emphasize a previous point. Pi is not a quantity. Whatever quantity we choose to assign to it is going to be arbitrary as to its precision. That precision is determined only by the contexts of its best usage, which are various and variable. No other assignment is sensible or even feasible.
What then of the idea of Pi; is that not fixed? No, again. It is certainly not fixed in our minds. Its meaning evolves with its usage and expanding relations to other entities. But is there not some final, teleological form of Pi to which we may refer? In the teleological context, Pi becomes embedded as an actor, or an aspect of cosmic intelligence, in the metanarrative, even as one among the Six. This is not something to which we could presently refer in any mathematical sense.
Numbers have both qualitative and quantitative aspects, and each aspect evolves toward a greater coherence. The fate of every aspect of Creation is increasingly seen as intertwined. Pi must partake of that teleology. Thereby does its normativity and functionality come to the fore. It can hardly be accidental that we come to know Pi in this context. Pi, along with everything else, exists to be known as a significant and integral part of the BPW. It will turn out the the meaning and function of Pi will be the best that we could possibly imagine.
Since Pi is such a key player and so relationally entangled, we can expect that there will not be much slack in its functionality. So although it has an aspect of freedom, that aspect will not be a significant feature, relative to some other entities. However, we can think of Pi as being the primordial prima donna of all mathematical entities. From that vantage, its intelligence would have had much wider scope, even freer reign, as it spun the web of mathematics and even physics about itself. I offer this as just a very crude and preliminary vignette. The pi of modern thought is a very pale shadow of its complete metaphysical self. If it still has any substantiality at all, that is its source. Being the measure of circumference may turn out to be an almost incidental aspect of its greater functionality and meaning.
Pi is also turning out to be our tool of choice in taking on the fortress of scientific materialism. It would be ironic if, in the future, we come to think of the function of Pi as more akin to an eschatological Trojan Horse than to mere circumference. It may be the string we have to pull to unravel this piñata that we like to call the world. But, on second thought, if it were our aim to depants the world of scientific materialism, what better item on which to pull, than its belt?
Having made some inroads with regard to Pi, our next item of the Six could be the Atom, but for pedagogical specificity, permit me to temporarily substitute the electron. What does Pi have to tell us about the nature of the electron?
As Pi is hardly just a quantity, the electron is hardly just a hunk of matter. From almost any vantage, it takes on the shape of a formal, ideal, functional entity; very much like its mathematical counterpart. Note carefully that physicists choose to refer to this entity with the definite rather than the indefinite article. Should this not, from the very start, raise a red flag concerning the materiality of the Electron? Where does the Electron leave off, and mere electrons take over. That ontological issue remains entirely mute. And according to one, almost mainstream, theory there is only one electron anyway. We are just seeing the same one shuttling back and forth in time.
As with Pi, how can we avoid considering its primordial, anthropically oriented progenitor? What was original with it was simply the sense of metabolic functionality. With metabolism there would have to be some form of chemistry. Energy bearing, interchangeable particles would be required. Stability would require that the energy levels be quantized in some fashion. This is precisely where e^i*pi enters the picture. Atomic stability comes from the mathematical symmetry of the electron wave functions. And that statement begs many a metaphysical question. How could electrons wrap themselves in mathematical symmetry, if they were not of the same stuff? And, indeed, they are, according to the super-symmetric theory of elementary particles. Yes, Mam, its mathematical symmetry all the way down. And how better to connote that symmetry than with with our circular Pi? Indeed the lowest energy level of the simplest systems is the shape of a perfect sphere, the only such instance in nature. It is the vast array of Pi's relations which has led us from circular symmetry to elliptic and modular symmetry all the way to the monster group. Plato would roll over in his grave if he knew that his ideal forms inhabited every speck of dirt piled thereon.
The elementary particles might be considered as spin-offs of the multifarious structures latent in the functional promiscuousness of PI. Is there a hint of moral turpitude in the reflected profligacy of our Matrix? Well, she is not called Mother for nothing.
[10/8]
OK, it is a bit of a trick to get the whole world out of one number, as seems to be the gist of the last few day's scribbles. But, in a holographic world, no matter where you look, you will see the implications of everything else. My pedagogical strategy for getting a handle on this convoluted situation is to single out a few special pegs on which, hopefully, we will be able to hang the rest of Creation. And, with even more luck, we may thereby hit upon a possible strategy for Creation. In focusing especially on Pi, I am attempting to co-opt the venerable Pythagorean tradition, which remains very much alive in mathematical physics. And replacing the Monster Group with Pi is helpful from both a functional and aesthetic point of view.
In all candor, the power of Pythagoras remains a principle cosmic enigma. I do not expect to be able to slice this world knot with a single stroke, rather I am looking for a strategy to unravel it in piecemeal fashion. The idea of the Millennium is to afford us ample time to complete this task, both theoretically and practically. Thus do we tame the eschaton, to whatever extent that is desirable and feasible. All I'm trying to do here is get this ball rolling. I just hope I'm not trying to roll it up hill.
There is ample evidence this ball is already rolling at a fair clip. In googling on 'world knot' I came across a book of that title by David Ray Griffin, and then discovered that he is a principle contributor to the Counterbalance Meta-Library. This used to be the Meta-Reference Library. It explores the boundary between science and religion by collecting the works and ideas of various intellectuals on the topic. They do hit hard on the Anthropic Principle; this is their strongest suit. Otherwise they are lukewarm, an occupational hazard of academia. CML follows the very moderate line of the American Scientific Affiliation, an organization of mainly Christian oriented scientists, who otherwise choose to remain in the mainstream of their professions. Although the ASA takes on board much of the Intelligent Design philosophy, it eschews their more confrontational tactics. And if you really want to get down and dirty, then just head on over to Creationism. In as much as the ASA has an official philosophy, it would be along the lines of Alvin Plantinga and Process. Process, in its turn, is a lukewarm idealism, which also means it is incoherent. It attempts to combine scientific cosmology with idealism. This is an oxymoronically futile effort, but it is also quite conducive to gaining academic tenure: no small thing, in many circumstances. Unfortunately, I don't have the time to frequently peruse these climes, so if something radical were to pop up, I might have already missed it. I am hoping that someone will keep me current with this potentially important and potentially cultivatable and captivatable audience. Back to the business of immaterialist cosmogony.
When I speak of unraveling, I may sound like a rabid deconstructor. This may be in keeping with my eschatology, but keep in mind that I am primarily a reconstructor of God. That is our telos. The A&O circuit is just the 'recycling' of God. Yes, there is a linkage between the Gotterdammerung (Ragnarok) and Creation; a linkage which I have not systematically explored, other than in its obvious connection with the Christ event. While on this subject, the following has occurred to me in comparing the Norse and Persian (Zoroastrian) traditions: the Earth is supposed to be the locus of the battle of good and evil, but it is also a struggle between the God that was and the God to be, i.e. the Alpha and Omega aspects of God. Obviously this temporal polarity can be very confusing for the monotheistic sects. Trinitarianism might be seen as a partial resolution, while Unitarianism, and the like, are a deliberate attempt to ignore the problem.
Now with Pi on board with the Six, there is a clearer emphasis on circuits and cycles (&ff) (more cycles, and here, here, here, here, Here, etc.). Everything fits into that groove, more or less under the tutelage of Pi. Are we in a position, then, to attempt a serious back-engineering of the Creation? I hope so. We'll have to see.
I am placing a heavy metaphysical and ontogenetic burden on the circuit and cycle. Is it justified?
9/23/03