|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.