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.
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.
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?
<-- Prev Next -->
Bootstrap is a term that has appeared frequently on these pages. (In this last link I am using the Google site search, which necessarily includes all of comcast.net. I encourage anyone to make liberal use of this feature, but please note that my address is changing from http://mywebpages.comcast.net/dantsmith/ to http://home.comcast.net/~dantsmith/.) [6/18/04 - Please note that I often use ouroboros (ext. ref.) or ouroboric circuit as being nearly interchangeable with bootstrap. (Google still references the old address.)]
At the end of the previous page I was about to launch into a discussion of cycles, but upon further reflection I have decided to include the bootstrap in that discussion as an important addition. I would first like to review some other uses of this term, particularly in physics.
It was Geoffrey Chew who introduced this term to physics in 1968. Geoffrey's interesting anti-materialist attempt to eliminate the notion of elementary particles from physics, quickly ran afoul of the ascendancy of quark theory. From a purely technical perspective the theory attempted to use only the self-consistency and symmetry requirements of the quantum scattering matrix to derive the properties and interactions of all the strongly interacting particles. But from a more philosophical perspective, Geoffrey's vision included a strongly relational cosmology which included consciousness as a fundamental part of the universal bootstrap. Fritjof Capra's The Tao of Physics (1976) made liberal use of this idea to support a Zen based model of physics.
John Wheeler's Participatory Anthropic Principle (1983) carries the bootstrap idea further in the direction of the BPW hypothesis, but it maintains an otherwise physical cosmology in a curious dualism:
The early multiverse can perhaps be thought of as a massively parallel quantum computer which explored all of possibility-space until it was able to generate a living body, which became the habitation of an observing, sentient being. At that moment the multiverse collapsed into the actuality of that one alternative environment.
A more mainstream version of the bootstrap idea in physics refers to the duality of particles and forces in the inherent symmetry of Richard Feynman's diagrammatic formalism. Here the swapping of the space and time coordinates appears to switch the roles of forces and particles, especially if one is not too particular about the fermi-bose distinctions.
In general, duality principles (3,500 hits) play a very significant role in math and physics. There is an inherent dialectical aspect to them, which in its turn may suggest an underlying bootstrap process as we see with Hegel. It is along these lines that I would like to include the bootstrap idea in a generalized concept of circuits and cycles, especially when they are viewed from an ontogenetic perspective.
While on the subject of circuits and bootstraps, I can hardly avoid mentioning Doug Hofstadter's 'strange loop' (33,400 hits). Here is the Math World entry:
A phenomenon in which, whenever movement is made upwards or downwards through the levels of some hierarchical system, the system unexpectedly arrives back where it started. Hofstadter uses the strange loop as a paradigm in which to interpret paradoxes in logic (such as Grelling's paradox and Russell's paradox) and calls a system in which a strange loop appears a tangled hierarchy.
Douglas Hofstadter, in Gödel, Escher, Bach (1979) and several later books, uses this esoteric concept mainly in service to materialism. His intention was to resolve the mind-body problem, in favor of the material side of the dichotomy, using this device. Daniel Dennett, Consciousness Explained, collaborated with Hofstadter in Mind's I (1981).
The Godelian aspect of the strange loop is a non-starter for Doug's purpose of bootstrapping consciousness. Godel's self-referential arithmetic construction presupposes a self-aware interlocutor, but should not be construed as contributing to that awareness. Also it is strange that Doug discounts the quantum observer loop as having any relevance to his strange loop. His commitment to the classical computing paradigm seems to be a marriage of convenience.
From a more qualitative, less technical perspective, there are aspects of the level shifting, tangled hierarchy which name the larger issues that any systematic cosmology will have to address. There is also a digital version of Pythagoreanism here which warrants our attention. Such a scheme might also be called 'information mechanics', but Doug's version is irreducibly loopy. To this significant extent does Hofstadter depart from materialism and even physicalism.