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.
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.
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?
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.
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.
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.