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?