Permit me to set aside aesthetics momentarily, to deal with the construct of space. I need a link between the anthropic and aesthetic aspects of math. It has something to do with combinatorics. The combinatorial view of space has long been championed by Roger Penrose, in a solitary battle. His views were motivated by Mach and Leibniz. According to Mach, the conventional view of space depends on an infinite background of particles. What if we start off with a finite number, wondered Roger? Then you have to consider Leibniz, and employ a quantum spin network. Space is constructed in a purely combinatorial and relational fashion.
Yesterday I had been watching Brian Green's Elegant Universe. The hero of the movie is Ed Witten, the fellow who in 1994 rescued string theory with his M (monster) theory. It seems that Ed has now taken up the cause of Roger's twistors. I haven't yet seen whether Ed's interest is also Leibnizian. My cohomology theory is regrettably weak. Ed follows the diagrammatic path of Andrew Hodges.
I'm now looking at John Baez' discussion of how Euler's discovery that 1 + 2 + 3 +..... = -1/12 has led to modern string theory. I'm not sure I buy this, Leonhard, that is. This is also the formula that launched Srinivasa's career. What is the sense of this, we might wonder? John shows that the 26 dimensions of bosonic string theory is directly related to this result of -1/12. What is the aesthetic of obtaining the finite from the infinite? Is it not the aesthetic of the microcosm? This may be where the microcosm impinges upon the cosmos, or where the mutual alignment is realized. And now looking at Conformal Infinity, another of Roger's ideas. A good discussion of the philosophy of divergence is here. Euler's discovery hinges on the unappreciated strangeness of negative numbers.
I refer you to A note on divergent geometric series by Kevin Carmody. He points out that if
* x = 1 + a + a^2 + a^3 + .... then x/a = 1/a + 1 + a + a^2 + ..... = 1/a + x.
* Thus x = 1/(1-a), except for a = 0, 1.
* Then for a = 2: 1 + 2 + 4 + 8 + ... = -1.
This fact is used to store negative numbers in computers by using their two's complement. You can see why the ancients were so wary of these negatives.
Having studied mathematics for many years, I am surprised to be surprised by this so late in the game. It is turning out, however, that the substantiality of the world is based precisely on similar renormalizations of divergences. The deeper we get into mathematical physics, the more sophisticated become our renormalization schemes. The entire quantum apparatus might be viewed as just one such scheme. Now I'm reading John's Renormalization Made Easy. Also take a look at Pierre Cartier's, A mad day's work: from Grothendieck to Connes and Kontsevich - the evolution of concepts of space and symmetry, Bulletin of the AMS, 38 (2001). Alexander Grothendieck draws parallels between semantic and physical spaces. He pioneered the constructive view of spaces. Epistemology replaces ontology, in the limit. Perhaps he can help us to understand how space emerged or was projected from the Matrix, as an alternative to the pokatok scheme.
Somehow, atoms must serve as proxies for our episteme. Must we anthropomorphize atoms and numbers? This is merely an extension of animism or vitalism. It is pantheism, and so we see why the theists are wary. Atoms, like us are chips off the old block, but this can also lead us into materialism. Materialism is pantheism without the 'theo', and without the possibility of apokatastasis. The restitution of numbers seems more logistically feasible than that of atoms. Mathematical physics ought to help level the playing field. It ought to expedite the relativization or deconstruction of atoms and space. We do this using the music of the spheres. Atoms and numbers are a medium for the telos. There is the not so spontaneous symmetry breaking of the Matrix by the dialectic. There is the sundering of synaesthesia.
Then there is Dennis E. Hesseling's Gnomes in the fog: The reception of Brouwer's intuitionism in the 1920s, as reviewed by Edward Nelson. Edward speaks of the formalist 'enlightenment', soon to follow, in which all semantics is superceded by syntax, pure syntax being the requisite of axiomatics. This prescription underscores the degree to which real mathematics, especially as practiced by physicists, departs from the ideal of formalism. By that much it must be animated, if you catch my drift.
Mathematics consists of reasoning and computation, and of the two, computation is the more fundamental. Aristotle, Leibniz, Boole, and Hilbert each took a big step towards realizing the vision of reducing reasoning to computation.
Thanks, but no thanks! I wonder how many formalist 'gnomes' are still around. Does the analytic vision still survive on the margins of mathematics? The mainstream shies away from philosophy. Does computationalism still survive in the philosophy of mind? Haven't we gotten over that? Did we forget to tell the formalists? If you want to know about computationalism, just talk to Srini. He'll give you an earful! Long live semantics!
I'm still lacking the proper metaphor for mathematical physics. What is the attraction of pure math, and why does it spill over into physics? How does the math migrate from atoms to brains? Is it all built into our construction of space? It may have to do with Rene Thom and his 'catastrophes'. Atoms and organisms are just 'defects' of space, or, rather, space is the sum of all its defects. This is scant support for combinatorics. Must there not be multiple dialectics operating on the Matrix to produce a stable weave? Or is Z the only stability? What was the precursor to the atom? The primordial catastrophe might be the atomic knot, then the alchemy.
I am just learning about clathrates and warming (3,300 hits) from an op-ed piece in the Baltimore Sun by John Atcheson. The global extent of these methane trapping hydrates only began to be appreciated in the early 90's. They range from the arctic tundra to the deep ocean sediments, and contain at least double the known coal and oil reserves. A prior rapid release of these unstable methane deposits may have contributed to a sudden warming episode 55 million years ago: Methane hydrates stir tales of hope and hazard (Science News 11/9/96). We may have to double the magnitude and halve the time scale of the expected global warming. Even with the Internet, news can travel slowly.
Usually it is the space lattice that gives rise to the defects, but we may need to reverse that sequence. Space may be a crystallization or sublimation of defects arising from the Matrix. Could the ego be such a one? What can twistors tell us about this? Do we wish to have space-time emerge independently of the creator? That might be putting the cart before the horse, almost literally. Should we look at Z as some kind of knot?
For more on twistors see M Theory Visionists and R. Penrose, F. Hadrovich.
What we don't know is whether the Matrix is primarily analog or digital, i.e. continuous or combinatorial and discrete, in terms of whatever potentialities may emerge from it. In my thinking so far, Z and D are some of both. The answer to this has a bearing on the types of pre-geometry that may be anticipated. Might any of the tools of field theory, e.g. symmetry operations, be usefully applied to Z or D? We could ask the same questions of mental states. How far removed is the Matrix from the symmetry of space and atoms? Will the answer be mainly epistemic or ontic? How big a role will logic play? It favors discrete entities. How heavily does relationalism favor continuity? Cognitive science deals mainly with discrete functions. This may be true even for perceptual systems. Which do we prefer, digital or analog? We'll take both, if we can. This issue also touches upon free will. Do we want to gives numbers a foothold in the mind? We deal with the cause and effect of thoughts. Are numbers going to help or hinder the teleology of the mind? From person to atom may be too big of a step. The single cell is a logical intermediary. That makes it harder to trace any mathematical continuity. The best we can do at this point is to check back with Rene Thom and his topological singularities in biological phase space.
Presently I am reading Temple Grandin's Animals in Translation: Using the Mysteries of Autism to Decode Animal Behavior (2004). Ms. Grandin invites us to use her autism as a bridge from human to animal consciousness. Now all we need are bridges to atomic and zodiacal consciousness. Horse whispering is a natural talent for the autistic.
A problem we face is that if atoms do not partake of selfhood, we will not be able to relate to them. Every cognizable entity must do so, and there are none other. This should tell us something about universal function, telos and apokatastasis.
What then is the relation between the selves and the ideas they entertain? Episteme always implies mutuality. This is the essence of animism. This is Srini's secret. He has intercourse with numbers. He is a number whisperer. There are no numbers, only number. Egos are the atoms of God. Atoms are God's alter egos. There are only states, be they mental or atomic. Sam's stone is a degenerate, unformed self. Politics is alchemy. Primality (of numbers) is something functional and ultimately only probable. I am my ideas and my ideas are my facets. There is no other context. Scheherazade must die when her story runs out. That is our fate. Every moth has its flame. The autist flies too near the flame, and is touched. Would we, wont we, all. Hiroshima was touched. It brought down the gauntlet. The skids are greased. Prometheus and Siva, meet Pandora. This is a restatement of the chain of being and its ouroboric character.
If the the self didn't already exist we would say it was impossible, but nothing else can exist without it. However, there must be relative degrees of selfhood, but can any partial self exist without an ultimate self? Is it then incorrect to suppose that a primal self emerged from from the Matrix? The Matrix and primal self may have to be identified, which would force a reinterpretation of MDX&Z. Can there be a difference between the primal and ultimate selves? The A&O are already seen as just two facets of one being.
If God is eternal, and if Creation is an essential aspect of God, then Creation is eternal, but then calling it a 'creation' would be a misnomer. No? One might then suppose that Creation and the MG have the same provenance. It is the eternal Telos. Then we have to deal with the flow of time in relation to the eternal Presence. We may already be beyond the point of the maximum linearity of time. If Creation were not essentially eternal, it could not relate to the rest of the cosmos, or to the MG. If physics were not logically necessary, there could be no mathematical physics. How then may we restore creatorship to God? Much of this has to do with the puzzle of presence, which is also about direct perception. Is Mozart a creator? The value of a creation is related to its endurance thru time. Mozart then is a transcriber of the eternal. His composition is not unlike the discovery of the MG. The idea is not to appreciate God by deprecating Mozart. God is the realizer of eternity. Her presence underwrites Presence.
Perhaps God is simply a virtual attractor, which is also the Telos. This is the God of the eternal future, who is finally present. It is our temporal egos that struggle to relate to the eternal self. It is the prophetic tradition which attempts to bridge the gap. What is our equivalent of the MG? There are the genetic generators and the resultant structures.
Creation is a collective process of which God is the coordinator. But is not everything pre-coordinated? Nonetheless, our struggles are real, and are essential to the process. It is the tension of the suspension (bridge). Ours is the optimal tension. God is the glue. The tension and the dialectic are related. They are both involved with symmetry breaking. We need to determine how God, egos, cells and atoms participate in the tension/dialectic. And what about numbers? How do these various beings rise to their many varied occasions? How do we manage to exist all at about the same time and share in the same mutual presence? Or is that strictly illusional? Time must be derivative of presence. Time and space are simply the mediators of presence. They often override direct perception. The problem of the mutuality of the present arises only when we treat time quantitatively or as derivative, or view being as non-relational. Do numbers share in this problem? Lack of simultaneity is a problem for relativity theory. There is the related problem of the directionality of time that enters into entropy and into the collapse of the quantum wave functions.
The great preponderance of what we think of as reality is based only on our counterfactual expectations. How are these expectations constructed and maintained, presumably in the collective subliminal mind? Is this the ground of materialism, the source of mathematical physics? Or do atoms have a distinct subliminal realm? To what extent might our expectations be able to steer events, and when do they fail to do so? Find the telos of the tsunami that can override so many expectations. There are cycles and then there are 'relaxation oscillations.' Some things bend and some don't. There must always be fault lines.
By and large, reality manages to abide by our expectations. Where it does not, scientists work to revise our expectations. Falling stars are an example. The timing of aperiodic events is left open. One might wonder how our varied expectations are mediated. How is the global coherence maintained? We suppose that it is the atoms that do so. Or are the atoms merely proxies? It may be the decoherence that is harder to explain. Atoms could also be the foci of decoherence. There are very few free atoms. They come in bulk, with standard bulk properties, be it a gas, liquid or solid. Consider storms and volcanoes. They all manage to be idiosyncratic. Are atoms to blame? Do they have minds of their own? A storm ravages a beach. Are the grains of sand to blame? Grains of sand seem to qualify as neither bulk nor atomic. There are many instances of such interstitial phenomena, i.e. the interaction of otherwise categorical phenomena. With a purely atomic metaphysic there would be none such. Can we deploy proxy atoms without falling into dualism? What might be the logistical distinction between atoms and dust? Consider Brownian motion. From dust to dust. And what happens in between?
In her book on autism, Ms. Grandin points out that 'animals and autistic people don't have to be paying attention to something in order to see it.' Things just pop out at them. Humans are more likely to see only what they are expecting to see. Her conclusion is that despite all the appearances, our normal consciousness is mainly verbal in orientation. Vision is brought under the control of the 'left brain' (pp. 51ff). On pp. 55ff Ms. Grandin points to the neo- or associational cortex. Autistics and animals have much less of, or have less functionality in their associational areas. We end up seeing mainly our own generalities. Also, as a result, normal adults are much more subject to ambivalence than are animals, autistics and children (pp. 88ff) whose emotions tend to be more compartmentalized. We might wonder whether atoms and dust can be conflicted. Does this have something to do with the quantum? Atoms are Nature's way of resolving her conflicts, as are we with respect to God.
Indulge my infrequent optimism concerning the imminent resolution of the metaphysical problems of immaterialism and the BPW. Taking our cue from Ms. Grandin, let us suppose that matter is the means to the resolution of cosmic conflict.
Before venturing further, however, allow me to acknowledge another member of the idealism club. This is an excerpt from the NYTimes 1/4/05, "God (or Not), Physics and, of Course, Love: Scientists Take a Leap: Fourteen scientists ponder everything from string theory to true love:"
Cognitive scientist, University of California, Irvine; author, Visual Intelligence.
I believe that consciousness and its contents are all that exists. Space-time, matter and fields never were the fundamental denizens of the universe but have always been, from their beginning, among the humbler contents of consciousness, dependent on it for their very being.
The world of our daily experience - the world of tables, chairs, stars and people, with their attendant shapes, smells, feels and sounds - is a species-specific user interface to a realm far more complex, a realm whose essential character is conscious. It is unlikely that the contents of our interface in any way resemble that realm.
Indeed the usefulness of an interface requires, in general, that they do not. For the point of an interface, such as the Windows interface on a computer, is simplification and ease of use. We click icons because this is quicker and less prone to error than editing megabytes of software or toggling voltages in circuits.
Evolutionary pressures dictate that our species-specific interface, this world of our daily experience, should itself be a radical simplification, selected not for the exhaustive depiction of truth but for the mutable pragmatics of survival.
If this is right, if consciousness is fundamental, then we should not be surprised that, despite centuries of effort by the most brilliant of minds, there is as yet no physicalist theory of consciousness, no theory that explains how mindless matter or energy or fields could be, or cause, conscious experience.
Now from the first chapter of his book:
But wait. If you construct all you see, then since you see this book, you construct it as well. And if that's so, then why should you buy it and why should I get royalties? What right have I to copyright your construction?
This question has more than passing interest to me as an author, and I shall have to raise a distinction to rescue my royalties. [...]
The except from Amazon leaves us hanging, and I could not get the search function working to retrieve more pages. I would like to hear more.
Ms. Grandin points out that 'normal' humans are uniquely prone to ambivalence, apparently in the furtherance of our quest for coherence. I happily construct a cosmology on this simple observation. The doctrine of apokatastasis posits restitution in the future. Various fundamentalist eschatologies rather posit Armageddon and division at the Omega. Let me suggest that they have made the common mistake of transposing the Alpha and the Omega. The paradisiacal zodiacal circuit is broken. It requires atomic metabolism and the human proclivity toward concrescence and coherence to make restitution. We finally internalize the predator/prey dynamic.
There is the primordial rupture: felix culpa. It has something to do with the primal sacrifice and subsequent theophagy as reenacted in the X-event. Sexual dimorphism is one small part of this cosmogonic event. We have had to eat and, yes, fornicate our way out of our predicament. We have grown so acclimated to our via metabolis that we can scarcely contemplate its abeyance, not quite the same as abstinence. Are not our own cells quite beyond copulation? Now we partake as if there would be no tomorrow: not an unreasonable posture under the circumstances.
Does this help in our understanding of atoms? Where would we be in our miscegenation without our recombinant DNA? Please note that organicity does not imply homogeneity. It is about the rainbow coalition. DNA, for all it's worth, need be little more than the logical detritus or our morphogenetic proclivity. Logic and love are our only glue. We come to the point where pure imagination can bridge the remaining gap in our cosmic circuit. Behold the spark gap.
Logic begins with the primal 'mark of distinction'. We can come home again, but only via the Riemann Hypothesis. It has been a struggle. Nothing will be wasted, not even our squeals.
A critical mass was exceeded. We are the fallout from the chain reaction. Atoms are the retarded cores of the primal circuit.
Keeping with the spirit of the quantum, it could have been a 'measurement' event which precipitated the rupture. We may still be in the throes of realizing that event, of reifying the possibilities.
Another way to look at cosmogenesis is in terms of a phase change within the matrix being precipitated at some critical juncture. That precipitous phase change is Alpha, involving a separation. From Alpha up to Omega, the cosmic partition is internalized within the microcosmic egos. This gradual historic process of internalization leads up to the final restitution that is completed within a couple of generations. I doubt that we have yet entered this final stage, but I would expect that the Internet will play a significant role in the initial phase of the apokatastasis. The Y2X event will signal its commencement. This website is intended to facilitate that event.
Does this sound too mechanistic? Well, no point in having God do busy work.
Suppose we switch from beach to forest. Fallen leaves replace sand. The trees are each teleological, and so are the blades of grass in the lawn. How do these differ from the sandy beach? Only in the associated teleology.
With the forest we can still tell the trees, if not the leaves. On the new mown lawn, there are only blades and grains, very similar to the beach. How may we move seamlessly between the two?
But we can tell the forest for the trees. It and the lawn are essentially systems. But then there is the petrified forest turning into a beach. Whose system is that in our young earth scenario?
We need not inherit the hobgoblin of Plato and Descartes: that true ideas must be clear and distinct. This shibboleth is a procrustean bed for relational, systemic thought. However, we must not belittle the individuality of each and every blade of grass. Each is a microcosm like ourselves.
What of the individuality of the archaeopteryx? Did her heart never beat? I have said that it is we who personify nature. I may have to stick with that. It is we who individuate. We do that because we are individuated chips off the cosmic block. We do this to numbers, as none others can. Numerology is not something gratuitous, juxtaposed against mathematics. It is the heart of the matter.
How do personification and abstraction coexist for us? Are they not antinomies? There must be combinatorics involved, but there is also integrity. Integrity is no abstraction, it is our soul. This may be where we come back to Temple Grandin. She points to the links between animal and human consciousness that are adumbrated by autism. Is Ms. Grandin lacking in integrity? Certainly not in honesty. Her refreshing honesty shines through every sentence. Is she lacking in humanity or personality? Perhaps she is correct in her self-diagnosis of lacking in cosmic ambivalence or multi-valence? The personal and cosmic are two poles of the chain of being. The world is suspended betwixt them. They are the Alpha and Omega in a slightly shifted perspective. Music is a link. It ties into both our emotions and abstractions. This is something lacking in animals. Temple has little to say on this score. Pythagoras had more to say. Temple says that we and the whales are latecomers to the musical scene (p. 278).
Is there nothing abstract about emotions? Is there nothing emotional about numbers. What would Srini say? Is there not a music of the spheres? How great can be the difference between the mental lives of Wolfgang and Srini? How far can either be removed from God? Did not Jesus have perfect pitch when it came to the Golden Rule, and how far is that removed from the Golden Section? Abstraction, microcosm and our social nature come intact. These come together in our linguistic skills.