Unreasonable coherence (0 hits). Are we surprised? Only mildly. By contrast, unreasonable effectiveness has 1,740 hits. This stark contrast of counts indicates to me a curious unwillingness to move from one very promising idea to the next in what ought to be a logical progression. Let us not fear to tread where none have trod, at least not within range of Google. [For more background on this topic please see the pages listed under relational metaphysics.]
My takeoff point is the Coherence Theory of Truth (1,500 hits). But let's back up one more step: Quine & holism (1,700 hits) and see my Quine page and here, etc. It has been just 50 years since 'Two Dogmas of Empiricism'. (In the idiom of the 1930's, I am tempted to say that the interim has been a 'three dog night'. We'll see if we can't warm things up a bit.)
Despite all our efforts, linguistic, scientific, logical, political, etc., to chop up reality, everything still hangs together to a remarkable degree. What holds the world together? From whence cometh coherence? I have the temerity to suggest that coherence is not merely a human artifice, no more than is the mathematical coherence of physics.
I suggest that there is a strong correlation between the coherence of the world (175 hits) and the unity of consciousness (2,800 hits). If the unity of consciousness is not a total illusion, as many used to allege, then its source is something that we often refer to as the self. If there is indeed an irreducible, immaterial self, then we must look for an irreducible source for the self. And so, logically, we must consider the possibility of a world soul (12,000 hits) which would be the transpersonal source of the self and its cohering world. I submit that this logical progression is now and forever the rational basis of all theism.
The closest approach to a rational theism is to be found in Platonism, and in the closely related Thomism. Hegel made a last ditch, pre-Darwinian effort to resurrect a rational theism. It has fizzled. It is curious that nothing approaching Platonism is to be found further east. The crowning irony is that it is the pantheist traditions of the East which stress a monist metaphysics. The West, despite its rationalism, has been more friendly to dualist and pluralist ontologies. Rationality begins logically in the carving up of the world. If the carving up is not restrained, however, it ends in the reductionistic nihilism that has engulfed modern culture. At some point we have to confront the task of reconstruction. That task has no recourse other than in the logical progression leading to rational theism, as outlined above.
This is the one thesis that I nail to the door of Modernism.
Darwin, more than any other individual, is responsible for modern nihilism. Darwinism remains a powerful force in the world. It embodies a simple minded logic that is compelling to the point of coercion. Its logic leads inexorably to the graveyard of reason, if not of life itself. There is only one way out of this hole that we have dug ourselves into. That is my thesis.
Now I need to find a likely door for my post-it note. Google has a list of about 50 philosophy forums. About a dozen of these are likely prospects.
Peter Suber also has a list of philosophy forums on his Guide to Philosophy on the Internet. Here is a listing of online religion forums using Google, but I'm finding few if any that would be suitable. The coherence of the world listing yields little of substance. For coherentism (1,700 hits) there are a number of useful links. The standard alternative to coherentism is foundationalism.
I continue to be impressed by the Monster Group (see here). The best source for 'monstrous moonshine' is John Baez (also see Tony Smith). No other object in mathematics appears to function as a greater relational nexus or irreducible source of coherence than does the Monster. Thus in mathematics it plays a role similar to the self in relation to the mind in general. It is only somewhat speculative to say that the Monster is also the primary source of coherence in physics, in as much as it is likely to be intimately involved in any final theory. Once again, I point to the polarity between the cosmic self and the Monster (see also e.g. here). Where would the Creator be without the Monster? Who created It?? I wouldn't want to meet that person in a dark alley!
According to John (and see week 193), the octonions, which have a special quantum, projective role, also have a special role in the exceptional groups. The octonions might thus be a link between the Self and the Monster, given the 'high octane' (E8xE8) observational aspect of QM. In this same week, John also compares 'classical' and 'exceptional' beauty.
Comprehension is a universal value. We comprehend that which is coherent. There are two types of coherence: formal and substantive, or syntactic and semantic, or objective and subjective. Objective, syntactic or formal coherence is rule drive. Subjective coherence remains undefined, and is often considered to be a matter of aesthetics.
Coherence taken to its limit is cosmology, which correspondingly comes in two extreme flavors: scientific and theological. There is no doubt that scientific cosmology is a crowning human achievement. There is also little doubt that it is, by and large, an already completed endeavor.
The one very significant question facing scientific cosmology is the outcome of evolution. There is only the general consensus that in the long-run, life in this universe will cease. Of great practical and immediate concern to our own species is the technological prognosis. There are widely diverging prognoses, but again there are two easily recognized extremes: technophilic and technophobic.
And just at this point in our very brief synopsis of the human situation, we confront irony and even predicament or paradox. The issue concerns the 'limits to progress'. Here rises, potentially at least, a cognitive dissonance between science and teleology. How do we describe it?
From a purely parochial, and short term perspective there is no great issue. Science proposes, politics disposes. The deeper issue arises only when we ask the larger question. How far can or will technology go; or, again, what if any are the limits to progress?
The closest that science comes to providing an answer is to be found in its treatment of the Anthropic Principle. This principle deals with the suitability of the cosmos with respect to the evolution of life. To make a long story short, there is the considered opinion that this suitability is remarkable to a degree that warrants an explanation. The forthcoming scientific explanation invokes the notion of an ensemble universes. Implicit, however, in this simple proposition are deep issues of logic, probability, and even objectivity and reflexivity. With this caveat, the closest thing to a scientific prognosis is the doomsday hypothesis. The dissenting, 'scientific' view is to be found in Frank Tipler's Omega hypothesis.
The doomsday scenario reflects the non-teleological, scientific view that evolution is non-directional. There is no natural guarantee of 'progress'. What progress we see is only a reflection of an anthropic bias, and it must therefore be ephemeral on the cosmic scale. On the other hand, Frank sees in the quantum observational reflexivity a cosmic mandate for observers, and thus is lead to a strongly teleological and technophilic conclusion. A transhumanic, universal progress is wired into the laws of physics. In these wildly diverging prognoses lies a fundamental paradox of science: its own unnaturalness; its own unreasonable effectiveness. All of this lies behind the notion of 'unreasonable coherence'.
Science is content, and very candidly has no recourse, but to let this matter rest. At the end of the day, what can it say besides, Que sera, sera? I believe, however, that this fundamental agnosis will not stand. Science reflects an essential human urge toward gnosis. It is not in our nature to be left in the dark, especially not after coming this far in our quest for knowledge. The show must go on. Can we possibly restrain our powerful collective proclivity? A crucial historical observation to make at this intellectual juncture is that gnosis and religion have never been compatible. The last several centuries of antagonism between science and religion are but one chapter in a dramatic, historical dialectic. With that major caveat, let us continue our exposition of coherence.
Coherence would be much less unreasonable if we were to take a more substantive view of the world. I use the word 'substance' advisedly. Substance encapsulates the pre-scientific notion of substantiality. It hinges on the twin ideas of being and presence. It looks at existence from a non-abstract, phenomenological point of view. Science is the great abstractor. It has abstracted substance beyond all recognition. In science the pursuit of substance has devolved into an arcane mathematical discipline of theoretical physicists, in which the most likely being is the 'Monster group'. One does not have to be an unregenerate skeptic to imagine that something of being has been abstracted out. We can, however, take a very useful lesson from the Monster: substance is something irreducible, the Monster is the awesome epitome of mathematical irreducibility. I will go a step further: substance and being are the essence of irreducibility.
Permit me to elaborate briefly. Science attempted to analyze substance. Scientific materialism was the result. We are left with atoms in the void. But these are not just any old atoms, mind you. Standing behind the atoms is the Grand Unified Theory of physics, and behind that theory, as just noted, appears the Monster group. Comprehending the Monster is a task for only the most accomplished mathematicians. It is the epitome of irreducibility, yes, and it is more. It is also, and even more strangely, almost paradoxically, emerging as the epitome of mathematical coherence. This last observation remains controversial even amongst its connoisseurs. I look to it not as a foundation, but merely as a potential pointer to a far greater coherence, that is truly the foundation of being. The Monster poses as the Gordian knot of mathematics. Many lesser puzzles of that discipline hinge upon its convoluted structure. This is decidedly contrary to common sense. On would expect that such a 'freakish' entity as the Monster would be self-consigned to a mathematical hinterland, but such seems remarkably not to be the case. It seems more to stand as the nexus of manifold complexity. But we need not and should not dwell on mere speculation. We have more substantive issues. Merely observe that motivation for aspects of our perennial gnosis can arrive from the most advanced fringes of the scientific gnosis. If we peer deep enough into the cosmos, are we not, fancifully, reputed to see the backs of our own heads, or at least our own tracks? What kind of reflection do we see in the 'Monster' that holds together the atoms and lends its coherence to the physical world?
My proposal for substance is at once radical and logical. Certainly it is phenomenological. In a phrase: to be is to relate. Existence is purely contextual. To posit an unknowable, unobservable existence or noumenon is to take the illogical, ill founded step of reifying the inconceivable. This proposed relationalism is tantamount to coherentism: all relations are internal or essential to the being in question. This is antithetical to what superficially is the scientific downplaying of relations by taking them to be external or accidental.
Let us consider further the contrast between internal and external relations. It was Isaac Newton's absolute reification of space that transformed what, up to then, had been a much more organic view of reality. Living as we are in the 'space age', it is at least as difficult to conceive of a non-absolute space as it was for Einstein's contemporaries to conceive of the curved space of general relativity. In particular, the odd notion of being lost in space was inconceivable to anyone before Earth was displaced from the center of the world. But even the now seemingly quaint geocentric view was a wild abstraction for the pre-Columbian mind. For the Aristotelian mind, there was not space, but rather places. Everything had its proper place in the world. For things to be misplaced in the world was a symptom of organic distress, disorientation and disease. Alienation was not a concept that would have sprung readily to mind.
Am I now suggesting that Copernicus and Newton were wrong? That Giordano Bruno got his just desserts?! No, but I am sure that there will be those who will want to caricature a postmodern, relational sense of space in this pre-modern fashion.
I have already suggested that, with the Columbia disaster (truly the 'displacement of a star'), we are witnessing the end of the space age. No longer can space hold the personal promise and fascination that it once held for those of us who came of age with Sputnik. Coming to an end is external exploration. The pundits may lament an 'inward turning', but was not the essential vision of the space age not just the Gaian vision of the Spaceship Earth? Ironically, it is ecology that is our psychic inheritance of the space race. Our burgeoning sense of ecosystems and general systems portends our postmodern conception of an organic world. Thus does space return to its more proprioceptive, functional sensibility. The Internet plays no small part in this conceptual shift. The essential metaphor of postmodern space is just cyberspace.
This might sound like the end of the story, but it is a bare beginning.
The antithesis of coherence is reductionism. Reductionism is our legacy from scientific materialism. The first step of our long journey back to coherence will be to roll back reductionism. This first step has already be taken countless times in the last half century, but for some reason the lesson has not sunk in. Reductionism is still considered a live option by far too many of us. Reductionism seemingly has nine lives. I would contend that it is already on its ninth life. Is there not just one more straw to find in order to break its back?
No, that will not be my strategy. That is the implicit rationale of postmodernism. Postmodernism seemingly wants to keep all options open, including that of reduction. Its vast stockpile of irreducibles are like so many Lilliputians against the Gulliver that is the residual coherence of materialism. Reductionism may be contained, but it is still very much kicking. It is not more Lilliputians that we need. What we need is a wooden stake.
Coherence is a vastly underestimated force. Or perhaps not. Postmodernism may only be truly understood as a loose but resolute coalition against coherence. Whence the resolve? Whence the intransigence? Like all intransigence, it is founded in fear. And, yes, fear of coherence.
The only way I know, or any one knows, how to deal with fear is to confront it. Naturally, we are doing anything but. Instead, on the few occasions where the issue of coherence has surfaced, there is denial backed by scorn, etc. The fear remains inchoate.
What gives? In the 'good old days' we gnostics were simply dispatched at the stake. No muss, no fuss. Now it is not quite so easy. Nonetheless, the cacophony of postmodernity has served to keep the ambient noise level well above the threshold of any nascent vision of coherence. That calculus, however, disregarded the Internet. The simple logistics of the Internet provides an almost clear channel for coherence. The ball is in the court of coherence. A little bit can go a long way in cyberspace. In cyberspace the shortest distance between two points is the coherent line. The self-organizing phase change from chaos to coherence should be virtually spontaneous. A butterfly flaps a wing and the currents of coherence are set in motion. The butterfly is a virtual, virtuous handful of truth seekers who overcome their fear. If there is a comprehensive, comprehensible truth out there, and half a dozen people are able to discern it, it will be all over but the shouting, thanks to WWW. Chaos and ignorance are on notice.
So, yes, for the last fifty years, the very worst kept secret is that there is more under the sun than 'atoms in the void'. That materialists now call themselves (non-reductive) 'naturalists' serves as exhibit A. The final guise of reductionism is not to deny the undeniable plethora of irreducibles, but rather to deny their coherence. The reductionists' cooption of the label of 'naturalism' is as explicit as they need to be concerning their fears of whatever monstrosity might conceivably exist beyond mere nature. For instance, we have already met the 'Monster', and I maintain that it is but a shadow. Whose shadow? Ultimately it is our own collective shadow, and that is quite naturally our greatest fear. Aren't all of our fears 'just' the projection of our own inner darkness? Yes, and we have ample, justified reason to be sore afraid of our own demons, collectively multiplied and magnified.
Indulge me, however, while I say a few more words while on this subject. The original and residual strength of reductionism was never predicated on its merits. In over a century now, there has not been one single successful, uncontestable instance of anything ever having been reduced to any other thing. The closest such case, and the one most often cited, has been that of thermodynamics being reduced to molecular kinetics. Let me just say that thermodynamics contains mysteries that remain at the forefront of physics research. The directionality of time and the loss of information are still among the greatest puzzles in physics.
The real problem lies with non-reductionism. By the same token as above, I can point to nothing that is incontestably irreducible. I submit that these twin failures are two sides of the same coin. The coin of the realm, the only legal tender is coherence. The substantiality of coherence is unanalyzable. You cannot, to the great frustration of the atomists, carve out a piece of coherence and place it under a microscope. In contrast, one can, famously, manipulate atoms under a microscope to spell out 'IBM', proving the reality of atoms? True, but only if physics can be identified with metaphysics, which is the whole issue from the start. Should theologians be surprised that God has not appeared in our telescopes?
Thus do the reductionists think they have succeeded when they place any, more or less tangible, piece of reality in their analyzer and proclaim that after exhaustive analysis that there is nothing left but tinier bits of reality. That is true, and it is notoriously true of the analysis of biological organisms, but there is a catch. The analysand never survives the analysis.
The portent of coherence lies in the very 'success' of analysis. The more we analyze, the greater is the 'complexity' that we discover. I only need aver that complexity is in the eye of the beholder. In the realm of complexity, the barrier between the epistemic and ontic becomes fully transparent, or should I say, totally permeable. Complexity theory is, on its own merit, the antithesis of reduction.
The reductionists claim the self to be an illusion. I agree. But, it is only a relative illusion. It is only an illusion relative to the reality of the cosmic monad. Otherwise, our own self is the most enduring reality we can possibly experience. It is simply the one nexus of all our experience and knowledge. Remove that nexus and what do you have? One blathering idiot. In this age of alienation, our lives all too often seem to disintegrate before our eyes, 'in real time', as we are wont to say. My only claim is that what little of coherence we are able to salvage from our lives can only be attributed to our beleaguered personal monad. The intellect is not everything, but it stands to reason that an intellectual appreciation of the greater coherence of the world would serve to ameliorate our alienation from it. Who knows, an integration of the psyche might be taken to heart.
I am just saying that coherence, if given any benefit of any doubt, can easily seem to overflow the strict confines of scientific cosmology. And I am not here to beat you over the head with coherence, but I do ask you, if you were never to give coherence a chance, how would you know what you had missed?
The gossamer of coherence lies in the ubiquity of relations. No one will vouchsafe that relations are all that we can ever know of the world. Things are merely a rational construct of their interactions with other things in general, and their interactions with us in particular. The reductionist maintains that relations are all spatial and temporal accidents. The coherentist demurs. Space and time, she avers, are merely abstracted from the essential, functional relations that comprise reality. The coherentist is a functionalist. We can never know anything outside of a larger functional, systemic context. Even, or especially, in a mechanistic context, nothing is left to chance. The place of everything is preordained in a cosmic causal context. The only idea that the coherentist needs bring to the mechanist's feast is that the universe may be more like a great thought than a great machine. Certainly, most, if not all, of the gears of Newton's clockwork universe have long since been replaced by the mental constructs of mathematical physicists.
From the view of developmental psychology, space and time are hardly innate concepts, but only develop very gradually out of conceptual generalizations of our functional, proprioceptive interactions. Contemplate the incomprehensibility of a simple map to a pre-modern mind. The Cartesian division of reality into the extensional and intensional or intentional is a nearly incredible conceptual leap, yet materialism is its legacy. How strange that our modern sense of substantiality came to be founded upon such a leap of the mind. How much stranger that we can hardly, anymore, grasp its utter strangeness. How do we find our way to disentangle ourselves from our own abstractions whose once very fashionable functionality we have already outlived? It will take some getting used to. Let us not, please, allow ourselves to be ground down by our own abstractions.
Enough of the fluff, right? Let's do the nitty and the gritty. What holds the stars in the sky? Where is Atlas when we need him? And how about mother Nature?
I take atoms seriously, but just not quite as seriously as my former physics colleagues. And I take the Monster group more seriously than most mathematicians. So here's the deal:
Ultimately I blame it (the starry sky, etc.) on logic, more than on God. There is no point in overworking God. Let's let God off the hook, as much as possible. In the BPW, there are ample reasons for a minimalist take on the big girl upstairs. Sua sponte, as they used to say in grammar school: it's up to us. I hold nothing against Mom, but let us not always hide behind her skirts.
Our two points of departure are Anthropics and the Quantum, and don't they just about amount to the same piece of logic, anyway? It all comes back to Wheeler's tautology: no phenomenon is real unless observed. Reality, then, is, practically by definition, egocentric. I recognize that this sentiment flies in the face of modernism and even postmodernism. Where is the humility? Should I not apologize? Not on your life. Instead I bite the bullet, the messianic one, that is. The above logic compels me to go the full distance. Copernicus was big. Whoever can overthrow Copernicus will be bigger, and we're talking biblical proportions. You get the picture. The logical coherence of our fin de siecle context demands that our would-be Copernicus II give obeisance only to JC. But that is not all. Crank in the previous observations concerning a minimalist theology, and realize that the 'Death of God' might be something more than a clever turn of phrase. Jesus was not the Son. Jesus was the sacrificial remnant. If that does not scare the Bejesus out of you, then, by God, you are not paying attention. Egocentric? Better call it Christocentric. In short: Copernicus2 = JC2. (I was always a big fan of indexicality!) The ultimate cosmic irony is that Jesus, in claiming to be merely the son of God, was committing a virtual prevarication, simply to help preserve our sanity and sense of cosmic security for another two millennia. But now the truth is out. Science has just been an acting out of this same cosmic charade, but not anymore, folks. It's no longer atoms. It's no longer God. It's just you and me, sister. And I'll do my messianic best to make that stick, right here at WWW. What do they say, 'Give it the ol' messianic try'? You and I know that this is mostly just a formality, but in this little formality lies a big chunk of history, and history is for keeps, after all is said and done.
Does any of this help to explain the starry sky? It is all part of the one big coherent picture. The best possible picture of the best possible world. There is, and will be no World2. The logic of the stars is simply teleo-logic, backed by the Monster. It is the Monster that disciplines our cosmic dream. We did not invent the Monster, nor it us, but we make the best of what we've got. The Monster might have been bigger, or maybe smaller. But where would we be without it? What if it were not the optimal size? Would this still be the BPW? Einstein wondered if God had any choice in creating the world. God wonders if Einstein had any choice in inventing general relativity. Just how coherent is our Monster friend?
As I have noted before, you can forget your pretty little starry sky, the Monster is the closest thing to an MIR that we are ever likely to encounter in this vale of tears. Close, yes, but no cigar. Nonetheless, it surely is the Cosmo-logical knot. There is a paradox in the combined mindfulness and mind independence of mathematical objects. How can something be so mindful and 'mindless' at the same time? The Monster is a stand-in for the final world enigma. Does it not cast its shadow on our little Millennial festivity here? Medusa could barely hold a candle to this mother. We might do well to employ the ruse of Perseus in avoiding eye contact. And where are those constructivists when we need them?
I have not yet declared defeat in my role of Perseus/Alexander/George vs. Gorgon/Gordian/dragon.
I remain desirous of resolving the apparent conflict between Anthropics and the Monster. Here are some points to consider:
- math genius
- number theory
- ellipses -> eccentricity -> modules -> exceptionals,
- TOE <-> Monster
- observer problems
- *numerical 'coincidences' (e.g. e^pi - pi ~= 20, etc.)
- 'organicity' of math
Are the above just a dustbin of mathematics, or something more? The numerical coincidences smell fishy. There is no explanation for most of them. The other points may or may not be related to this problem.
I want to show how the Monster might have been created, but is it not also nice to have a true MIR? There is ambivalence.
There were bound to be exceptional groups; and, perhaps logically, there was bound to be the largest one. These exceptionals would figure in the various mathematical exotica, and possibly in math genius. Primality is primal.
Math genius may pick up on the resonances of the near misses. With no miss, there is no resonance. Quantum computing could do likewise. There could, on further speculation, even be an optimal observer bootstrap loop; and that, of course, would be the crux of the matter. Could there be an irresistible force hereabouts to move our favorite immovable object?
There may be virtual, shadow, 'imaginary' monster-like 'resonances' (an infinite number of them?) of even larger dimension. We see now just the largest 'real' one. We're talking imaginary 'roots' of some larger system. It may be that our world exists 'off the mass-shell'. It is reified teleologically, and the reification of the monsters or exceptionals is related to that. More or less! This speaks to the putative organicity of mathematics. Math, as a purely mental construct, cannot be exempt from hylozoism. Can it? Could we conceive a logic incompatible with our existence?
There is an optimal size for the largest real exceptional group. That's the one we observe. That is no coincidence. That is a logical necessity. Logic is not absolute, nor is the resultant math. There is a 'gravitational' warping of the logic. Any extant logic space must be closed epistemologically. Math is not exempt from the necessary epistemic-ontic connections. This closure is just the foundation of coherence. This is the source of the 'unreasonable' coherence of the world.
How and where does this hylozoic 'warping' of logic operate? It operates on the fringes of our cognitive abilities, and it operates there with sufficient coherence to be virtually undetectable. It operates in the realm of genius, which remains the foundation of mathematical truth. It is the universal resonance of genius. It will be as elusive as vitalism to detect. Consider the dog that chases its own tail! It can only be seen coherently or holistically.
e^pi - pi ~= 20? Does this not just compute, mechanistically? Is the circle warped? Yes and no. Forgive me if I point out that 'e' is the number of life, and pi is the number of closure. How could there not be this near closure of life? Is this too spooky? (JC2 = 20C AD as the closure of history, but don't complain to me, I just work here!) This latter is only meant as semi-serious, but it does point to something truly serious: the organicity of the world and all its 'abstractions'.
Could we conceive of e^pi as anything else? How did we fudge these numbers? No, they fudge themselves. This is a true panpsychism, just as advertised. Is this not a gross overdetermination? And what else is new in the BPW? Even the numbers conspire, for they could do none other. The natural logarithm is a vital, organic concept. It must resonate, and e^(i*pi) is only the mother of all resonances. Better stop me before I commit any worse crime of numerology. But do recall the antagonism between the decimal and the duodecimal forms of 'closure', as apocryphally pointed out by the Jews to the heliophantic Egyptians upon the occasion of their exodus, with the first ever 'recorded' street gesture, and this despite their more traditional Davidic semiotics. Thus we do not have a mod of 24 as might have seemed more 'natural'. 10^10, it turns out, is more natural or organic than 12^12. The logic of numbers appears fixed just because of our fixation with it. C'est la vie, mon amour. Is Pythagoras turning in his grave? If he is, it is to a 5/6 rhythm.
I recognize that I am terribly far from the semblance of an explanation; but do, please, recall that this is an exercise in minimalist messianism, and so: sua sponte, my friend. Your work is cut out for you. I'm just the entertainer here. I'm the one with the squeeze box, when I last checked on Google.
I'm surprised to find only the above reference to the e^pi coincidence. It was not even reported until 1988, which seems rather late in the game. I can only conclude that mathematicians must find it somehow embarrassing. A less peculiar coincidence involving the sqrt (163) has at least 50 references, probably because there is an 'explanation' involving the 'j-functions' that are related to the Jacobi elliptic functions [see the Heegner numbers]. [3/11: I'm still not seeing the 'explanation'. Does anyone else?] [9/1: There is an ambiguity as to whether the e^pi coincidences determine the j-function or vice-versa. The 'rationality' of the 'j' is the issue. From whence does it come?]
I don't usually take up the Pomo cause, but here is an exception:
CONSENSUS AND COHERENCE IN MATHEMATICS - HOW CAN THEY BE EXPLAINED IN A CULTURALISTIC VIEW? by Susanne Prediger, PoME journal 16, July 2002
To posit mathematics as cultural, as Susanne virtually does, is simplistic. Yet how much less cultural than chess is it? I would venture that math is more 'cultural', given the greater rigidity and simplicity of the rules of chess. The concept of 'proof' is notoriously vague and controversial, as are the various axioms of logic. The various treatments of infinity and the calculus are cases in point. The role of conjecture is absolutely essential to its advancement.
Does any of this consign the Monster and e^pi to the status of artifact? Yes and no. Suppose we speak of a cosmic artifact? Does this present another picture? Is 'e' not an artifact of some culture? It emerges from the culture of calculus. It is an article of that culture's coherence. Is not 'pi' an epitome of some kind of culture? Both depend upon controvertible notions of infinity.
I would say further that mathematics has no choice but to be optimally coherent. We're talking FLT (Fermat) type coherence. We don't yet fully appreciate the psychic status of 'deep' math, any more than we do of deep ecology. Maybe we do see the back of our own head, after all -- no longer quite so 'fanciful' -- coherence is the only thing left unbounded (Universe as Doughnut: New Data, New Debate) [Dr. Spergel added, "If the universe were finite, then this would rule out inflation and require something new." [...] Moreover, the idea that dimensions could be curled in loops occurs naturally in theories that try to unite gravity and particle physics, several physicists pointed out. (Micro/macrocosm!) Anyone for coherence? All this harks back to the Greek abhorrence of the Apeiron.]. Until we understand the deep ecology of math, we cannot dismiss the 'coincidences' of e^pi. Numerology and astrology are not wanton. One could hardly imagine a culture without them. Coherence resides in those shadows. Was Ramanujan (1887-1920) not just the ultimate numerologist, much to the chagrin of many of his less well endowed colleagues?
Gosh, I was thinking I had already made ample reference to the Ram, but spell-check tells otherwise. He is surely exhibit A, but for what crime? Whodunit? Hedunit. There is a very peculiar resonance between our deep psyches and deep math. The three journals devoted to the ram-ifications of his thought processes are a small testimony to that....deep coherence? Should he not be ensconced in any prophetic pantheon? He is already in mine. The Ram has put the genie back in genius. Case closed? Math is no less optimally coherent than ecology. The BPW is the notice of a cosmic ecology, nay, a cosmic teleology. Gaia is written on that wall.
Can we Ramify Jesus? It is a similar idea, but different orders of magnitude and quality. It is also about being in the right place at the right time. Tell me about that!
Mathematicians are sitting on a metaphysical goldmine, or is it a time-bomb? They keep quiet about it lest they upset their hard-nosed, scientifically established colleagues. The unreasonable effectiveness of math is forcing the issue on the larger community. The larger community looks askance. The larger community dreams of the genome and then the proteome and the unknome raise their heads. Complexity knows no bounds, other than those of an ultimate coherence.
The deeper we look the more we confront an irreducibly organismic (unreasonable?) coherence in math, physics, psychology, biology, etc. Where is the epistemic/ontic divide? When will we admit there can be none?
Have I slain the Monster? No, but I have tamed it. To everyone's satisfaction? Not quite yet.
Does the rational theist wish to tame God? Do I wish to place God in the traces of our telic ambition? The cosmic self has a mind of its own. We would not be here to extol it, had it not. God will be tamed only to the optimal degree. And whose optimum is that? That is just love's maxim. And if you think love will be tamed, well, do I have news for you! This truly is news from nowhere and now here.
Back to Susanne:
In short: coherence in mathematics emerges, because mathematicians immediately search for solutions to level inconsistencies whenever they appear. In consequence, inconsistencies do not exist in mathematics, because they are not tolerated.
By the way, 'coherence in mathematics' yields only 31 hits, and all but this one deal only with math education. Unreasonable coherence is unreasonably unattended.
The Pomoists miss the bigger picture. They scarcely distinguish mathematics as one among many cultural artifacts. I suggest its role is crucial. It is the most direct probe of the universal psyche. Indeed it points to that universality, as nothing else can, abstractly and outside of religion. Am I positing the quantitative nature of the psyche? No. I am positing the unique, qualitative nature of mathematics. I posit the emergence of sheer quality out of sheer complexity. Complexity is the Rorschach, mathematics is the test. Mathematics is the end of complexity. The Riemann hypothesis and the classification theorem point to that end. There is a closure to complexity. The unboundedness of love is a singularity unto itself; the rest of the world is its shadow. Mathematics circumscribes that shadow as nothing else can.
Ultimately, of course, coherence cannot be analyzed. Mathematics comes as close as possible. Beyond that, coherence can and will be celebrated. It will be lived. It is our magic carpet: the genie of our genius and genus.
The unreasonable effectiveness of mathematics is owed to its logical circumscription of chaos, of the Apeiron. We see its tracks in the micro/macrocosm. It is the testament to the logical confluence of ontology and epistemology. Mathematics has tamed the noumenon and placed it in the traces of love. Sorry, but there is no more analytical way of stating it.
From whence comes math?
Consider all the mathophobes. Is math really necessary? Couldn't the BPW be mathless? I can sympathize. I consider math to be a sometimes entertaining spectator sport, along the lines of chess, but that does not mean I would ever want to spend more than a few minutes at a game.
I submit that, indeed, it is games that are the ontological source of math in the world. Why games? They are entertaining. It is often hard to distinguish between life and sheer gamesmanship. Darwin took this basic insight and turned it into a reductio. We need not go there.
Games entail rules, and rules entail logic and randomness generators. Where would the game of life be without the laws of physics? Metabolism is a major part of that game.
It may be that in heaven there is no metabolism and no physics to speak of. Why not just skip all the restrictions under which we operate here? It comes back to spiritual gravity. There has to be a place where the risks and stakes are high. Again, there is no way to avoid this sort of game talk. Heaven may defy gravity, but it has to wait upon the game down here. Otherwise the world would not be serious. It would resemble a joke, even more than some claim it does now. There is something about truth and consequences that we cannot ignore.
There is another possible consideration relative to the necessity of math. In the quantum world, math enters in a more fundamental fashion than in the Newtonian world. This has to do with the discreteness of observations and the entailed probability distributions. The stability of matter depends on these rules of observation. If there were not a rule concerning minimum photon energies, the world would implode in a microsecond, a very warm microsecond. If there is to be a coherent microphysics, quantum-type rules will come into play, and so will the attendant mathematization.
With quantum systems, one has to deal more directly with system states. Such states have formal observational classification schemes, which entail sophisticated mathematical structures, and there is no getting around it. Such schemes do have game theoretic overtones, although I am not aware of any formal analysis of this nature.
In classical mechanics, the formalization is something that may be appended to the underlying dynamics, often in an arbitrary fashion. With quantum physics, however, the formal aspects are right there in your face. There is no way to define such a system without being explicit about the mathematical machinery. With the quantum realm, the epistemic-ontic divide is explicitly overturned, and it is done so in a necessarily rule driven, mathematical, computational fashion. Quantum logic is virtually a pleonasm, whilst Newtonian logic borders on the oxymoronic.
The rule driven nature of games makes them eminently suitable to computerization, up to and including the simulation of the contestants themselves. But how do we get from the normative character of games to the inviolable nature of physics? Probabilities likely play a role in this transition. And what plays the role of the computer in this informational mechanics? How are the rules enforced?
There is something about the natural discreteness of information that should be considered. Is information more naturally digital than analog?
To come to grips with my questions about the place of mathematics in the natural order, I need to appeal to creation. Is there any logical order of emergence outside of the Darwinian model?
Perhaps the most primitive of all systems is just the cycle. I would submit that the lineal logically emerged from the cyclical. The myths of cycles appear to be the most primitive. The cycle combines key semantic, semiotic, physical and mathematical concepts. It is a root dynamic. A crucial aspect of the cycle is the resonance, which combines space and time in fundamental fashion.
In considering the semiotics of resonance one can hardly miss the nearly universal presence of OM:
OM, like Amen and Ahmeen means the Divine which manifests as all that is. OM is the "word" as in; "in the beginning was the word" which means primal sound or vibration from which all other names and forms arise and to which we all return. OM is the Alpha and the Omega.
In theses senses, OM is logically prior to the more lineal prophetic logos, or the Zim-zum of the Kabbalah.
Pi is the measure of the cycle, but, a much more self-contained and dynamical representation of the cycle is e^i*pi. The imaginary unit (i = sqrt(-1)) is a resonance within a resonance. e, i, and pi are a virtual trinity for the discipline of mathematics, not to mention for physics, and especially for quantum physics. No other quantities can match their power or ubiquity in the most basic and esoteric realms of math and physics. The cycle cannot be considered to exist independent of these quantities. They exist together at the most primitive of all ontological levels. If there is a confluence of psyche and physis, mind and matter, or quality and quantity, it would be here.
It is the magic of the Zim-zum or lineal logos to break, temporarily, out of the cycle. That temporality becomes our history. Six is the circular number and five is the lineal number. I would equate the pentagram with the Zim-zum. The peculiar decimal syzygy of e^pi - pi = 19.999099979... cannot be ignored in any such context. If this syzygy does not point to the breaking-out nature of the Zim-zum, then someone is trying hard to fool us. There is a semantic necessity here which overrides mere numerical necessity in a manner which is reminiscent of the Anthropic role of the Monster group. The coherence of mathematics and the world hang precisely in such a delicate balance. Is any of this unreasonable? No more so than the coherence to which it gives rise. The implicit numerology is just a minimalist version of panpsychism. At the end of the day, numbers cannot and will not alienate themselves from the mind from which they arise. One can no more remove numbers from the mind than one can remove mentation from the numbers. Should this logical reciprocity not have been more obvious to us? What mental block led to the modern epistemic-ontic divide, especially with regard to numbers, of all things?
There is in the world a ubiquity of cyclical phenomena. The numerical quality of cycles is essential to them. The cycle is apriori irreducible, both quantitatively and qualitatively. Nowhere are the epistemic and ontic more closely conjoined. The simple cycle should have been a sign to those who so glibly sought to put asunder that which is logically indissoluble.
Can there not be cycles beyond our ken? Does that not depend on 'our'? The notion of a cycle is conflated with the notion of self-identity. Most everything else is trivially or tautologically self-identical, but not a cycle. Its self-identity is non-trivial and essential to its existence as an irreducible, emergent property. One such property is frequency. Frequency is not an arbitrary measure. Frequency is essentially and physically relative. Frequency is essentially observational. Could we not say the same of extension or duration? I submit that frequency is more basic than either. This fact is formally recognized in the designation of the metric standard in terms of wavelengths of light. Length and duration are both derivative relative to a standard frequency. The absolute nature of the velocity of light speaks to this same logical and physical relativity. And through Einstein, even mass is reduced to frequency. The reduction of energy to frequency forms the basis of quantum physics.
Well, gentle folk, you may have wondered at my absence. Among other things, I have been wondering whether, if I were to be shot, the hole in my head would be caused by the bullet, or would be psychosomatic, and how I would know the difference. This might seem a far cry from the e/pi syzygy, and it is. I think it is easier to explain the starry sky, given the likely necessity of a sunny sky, than a hole in the head. Cycles are certainly more obviously involved with Sun and stars than with bullets.
I have taken about 100K of notes on cycles and stuff, off-line on my PDA, but I have nothing that is sufficiently succinct for these pages, yet. To wit:
Friday, April 11, 2003
· creation may be like rebuilding a society such as Iraq
· a society is composed of ring structures built out of ring psyches as a magnet is built out of individual spin atoms
· the will is the direction of the spin
· alignment is the problem
· cloning of spins, and then a reproduction cycle
· social space v. phys space
· sexual repro is a trick
· that is about the biggest leap from the pure psychic realm
· what is the prototype?
· repro has a lot to do w/ atoms
· can we do a Feynman ring diagram for repro, and have it connect w/ atoms?
· s-repro is a way to prevent cloning
· what is the primal prototype?
· primal seeds?
See! Take my word for it. Talk about bullets....
A problem that is inverse to the bullet hole problem is that of walking, or not walking, through walls, which reminds me of the 'bullet hole house' in Concord, but that is another story. Normality is more of a challenge for immaterialism than is paranormality. They don't shoot ghosts, do they? How do we invoke the Pauli Exclusion Principle, without invoking atoms? I don't mind atoms too much, we can derive them from cells, as cells are derived from organisms, given some sort of scheme reminiscent of 'object oriented programming' with inheritance and the like. But is that too much of a stretch? Speaking of stretches, how many people are aware that the PEP depends on fermi statistics, which, in their turn, depend upon, I believe, something like PCT symmetry, and don't ask me what that derives from? Sometimes physicality is not quite as physical as some of us might like to think. Nothing like a little physics to disabuse one of physicalism.
Looking at it from another perspective, we normally avoid the superposition of heads and bullets, but doing so is usually related to the problem of Schrodinger's Cat. Who observes the hole in my head? Is it the same Gal who observes Berkley's tree on the quad? And how does the shooting of bullets relate to the chucking of spears? How many spears would a....? I don't mean to make lite of the silver bullet-hole problem, but what else can a body do?
When I get serious, imagine that, I get serious about the continuity of cause and effect. Given the A&O and the BPW, everything else has to fall into place, a la the PSR. The manifold of reason is quite continuous. You can stretch it out, and the Lord knows we do, but you can't break it, anymore than you can break a gluon string. Can we not work the superposition problem as well from the top-down as from the bottom-up? Perhaps even better if we have to invoke the indefinable notion of observations and record keeping. It might be easy to make a record, but not so easy to keep one. Just ask the IRS, but please don't tell them I sent you.
Reasons are manifold. Less often are they manifest. Does not a reasonable world become grossly overdetermined? But that overdetermination is just what keeps the roulette wheels in Monte Carlo honest, honest! Sure there is psychokinetics, and there are also casino bouncers. Being discrete is not an option down here, it is the best possible law. I need the PSR like I need a hole in the head. We all have holes in our heads, it's just that they are hard to observe. Well, they would be easier to observe, if we weren't so darn persnickety about them! It's not necessarily God that keeps the trains running on time, it is more likely our own insecurity. That good ol' 'invisible hand'. And that is what the Omega is (not) about. It is just about homeland security, if you catch my drift. If you shoot me in the head I will die, but not out of necessity. Out of insecurity. It's like the evil eye, only more so. And if you ask me no more questions.... It must be time for another vacation, or another war. The telephones don't really work, its just the commercialization of telepathy, to slightly oversimplify the matter. Have you noticed that when the wife calls, the ringer is practically superfluous? It's only the telemarketers who really need phones. You mean you didn't know that communists don't use phones?! Shame on you! Or was it the Amish? Anyone for a little shoo-fly pie? How about a telepathically telemarketed shoo-fly pie? You see the problem, don't you? So, will the glory train be running on time? Yes, if it runs on Internet-Google time.
And what about SARS? Is it eschatological? Is it another embarrassment for immaterialists?
As you may have noticed, I am not easily embarrassed. Not by nature, just by profession. Will this be the Big One? How will it play in the hinterland? Will it genetically combine with HIV? Well, yes and no. Be scared, but fear not. Will we be robbed of our Millennium? What do you think!
There is talk that SARS may take down the Beijing regime as effectively as the Marines took down Baghdad. That would not terribly displease us, if we can contain the loss of blood, minimize the looting. It is possible that SARS could be restricted mainly to China, where there could still be millions of deaths, and perhaps Africa, where the HIV recombination would be almost foregone. There would at least be a behavioral transformation on the Sub-continent, if not in the Third-World. The cost is great. What is the reward? Is this the price of the Millennium? Are those the ashes from which the Millennium must rise? We will shape up or ship out.
Along with Beijing, Islam, as we know it, may be another casualty of SARS/HIV/terrorism. Kim Jong's days are numbered. He is the wrong person playing the wrong game at the wrong time. He might not last out the year.
What then of idealism? Will there be a break or a new opportunity in this game of global dominoes? Will there be a new crack in the pall of materialist orthodoxy? I can't yet place it.
And why SARS?
Could there be symbiosis without parasitism? Could there be biology without symbiosis? This is what ecology is about.
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