Tuesday, February 12, 2013
The Quasi-discrete and the Quasi-continuous in the Intuitive Symmetric Plane
L. Edgar Otto 11 February, 2013
Micro-tones of integer ratios with respect to just notation allow a merger with the chromatic twelve tone tempered system of music for consideration of the change of key theme hierarchies and this too is a method of understanding the intimate relation between abstract space and numbers.
But as regions of the quasic plane the discrete integer ratio values are quasifinitely expresses as square or quadratic forms in the number theory. The general idea of proof of Fermat's Last Theorem should not be an end of interest in the sub depths and spans of familiar low level dimensions and numbers. In particular and as a pattern parallel that is the basis for that proof the answer sought for by integer counting methods involves the abstract dimensions as represented, creating an abstract symmetry hidden to the dimension of plane flatland, between two and three space.
These are quasic as they are issues of quasi-contiguity. What may touch in some sub-space dimension may not necessarily do so in another sub-space dimension. The idea that what can be represent as triple integration adding or subtracting a dimension is the same as partial integration does not necessarily unify the descriptions between such contiguity as equivalent in some directed or evolving degrees of freedom to which physical functions may set the bounds of behavior.
As a very simple exploration, the actual counting of areas in such abstract quasifinite ratios, I looked at the comparison of actual pixel area count and methods related such as our ideas of dilitation or congruence. From a foundational view on that frontier it is not at all clear our intuitions unto this exploration will yield mroe than trivial results- these moreover to be abandoned as we strive to explore wider laws of mathematical physics beyond such trivialities of the Euclidean plane to which we regard then as too obsolete a view as to consider it progress in science to explore- in other words the research direction itself is an enterprise of quasifinite understanding and imagination as self limiting.
This deeper view of the plane region thus not only allows for the distinctions of symmetry (including chirality and parity) but is the foundation at singularities for asymmetry as well to which we may follow in some way the structure of our theories.
We know from more general considerations of space and number, expressed as questions or paradoxes, usually, as with any perplexity of a truth in our decisions of thought, that the whole (equal to its parts) does admit certain nodes in a closed orbit or in a recursive sequence (irrational even as a span) to which our ideas of complex numbers humanity has long computed and applied, even as sacred numbers in a vauge mysticism or worship.
We know that beyond a certain point so expressed as this mapping unto or at some infinity as complex planes or more general abstract brane multi-dimensional representations, the operational laws of mathematics run out before the continuity considered and endless dimensions.
In these imagined, (perhaps thus possible) conceptions of space and number operations, including possible new operations or symmetries imagined, we do not necessarily heed the speculations of the inventors of such methods as Quadratic Reciprocity that beyond a certain level of complexity the theory is not left comprehensive and complete. Modular numbers with residues go way beyond our simple programming algorithms of residues and so on.
We know that the real number plane R, can be further extended *R. So why not the depth of its differential geometry? Why not the consideration of a wider realm of p-adic numbers to which in regard to limits and the infinitesimal the Great Dedekind gave us a much deeper yet different view? Or in such abstract realms we may light upon the ideas of a higher vision, treating the irrationals as surreal in a calculus?
Other than our ideas of what is the standing out as if from a different dimension, the whole greater, equal, or less than its parts, this nature of insight that is the substance of insight, the business of physics is intimate to the binary yet vague or wild card considerations, thus the variety of numbers as in the power continuum as if subsets and the unique span of a continuum. In particular much of physics, as well the errors in measure or theory so resolved involves the half real in an imaginary number ( just as there are hyper-reals, there are hyper-numbers complex, up to nine dimensions in their calculation as imaginary...) To these various power continua the power raised say as a prime with hierarchies of powers or with certain integral differences on the same level we try to put physics into some side of its quasifinite number forms.
This is lately the postulated inclusion of so many numbers of holes that describe embedded or enumerated manifolds of geometry on the way to a trivial or unified theory free from our quaint ideas of a foundation really shown just views of one structure as in the first imagined enormity of string landscapes and compactified dimensions.
Notation itself may be extended formally yet have a use in the future application in the sciences. Coxeter's reflections on symmetry exhausting space up to 8 dimensions may be extended beyond this- so too Conway's notions of 24 dimensions, apparently both levels of insight still not allowing higher levels as possible in a one universe complete whole to which the application to physicality may develop partially in its methods we imagine valid but not reachable or provable unto some infinity as the universe diversely evolves.
In the counting of regions of the quasic plane we thus have to consider the sub-space grids and holes. In theory we may discover some of these as properties of particles, or we may acknowledge that some known particles do seem to have their mirrors in space and time while asymmetrically so and there exists at least half spin particles. Certainly, we see this general halving of things in paradox a part of the properties of topology even when the parts are jumbled. We see also the paradox of mapping any such abstract intrinsically flat line as if a continuum into a circle or a sphere. But one thing seems sure, local time has many subjective and objective variations. After all, in the impossible clock measure of time in the world it is clear that a stopped clock is right so many binary times a day.
If this research proves ultimately trivial, well, forgive me my mathematical recreations, it makes for the beginnings of some interesting digital drawings in perhaps an infinite variety of them...some of these ratios are pleasing in a sense of beauty I suppose for instance 89 is Sophie Germain prime as well as 23 and so on I consider as pixels. But in my art like drawings one should not assume all of them have a hidden sense of color or ratio, discrete or toward the square or hexagonal cube analog to decode- yet for the actual drawing if not the sketches toward an idea I have not other than some initial learning of skill on some level of method drawn thing related to science as if free brush arbitrary or coincidentally discovered order in the chaos. Where there is something to decode is it set in the patterns carefully.
Perhaps we are limited in many respects as to how far certain numbers may go such as the Mersenne primes so favored by Pitkanen and his number 89 and so on (I am still trying to sort out the parallels with his and my quasic theory) so we need to show why such limits- or not necessarily for when we know the patterns we do not need to compute all the possibilities to infinity in the inductions (or as mathematical inductions) beyond a few first levels to know on some level something like pairs of primes extends forever. But, let us hope some viewers enjoy the illustrations for the art's sake if nothing else.
Part of defining consciousness, by the way, is this creative process to which we express or desire out engagement with the world and discoveries for those who may care, that part of it made more than and less than in the empty whole, the substance of it as made of insight.
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Clearly, our stance to holes and the bifurcation in them amounts to a global logic- as to what happens in the brain.
This vaguely a higher structure of which we imagine emergent wholes different as defining intelligence that its parts, and with the idea of thresholds, that multiplicity of a single photon cascading others in retina stimulus or in photosynthesis as if a model of artificial intelligence, or as logic circuits the simple states of signal or no signal for a zero ground- logic as algorithm or these holes as non-algorithmic. But the higher level or lower level over indefinite space may have this distinction top down and bottom up that do not necessarily correspond. Nor does there need to be a material structure for such dynamics of space.
If the eye as part of the brain (Penrose) then the brain too has this process of merging or halving what we imagine as visual illusion in which to distinguish what is outside our quasifinite apparently closed surface as we contemplate what in the dynamics is separated space like in the speculations on relativity and black holes unto singularity.
We see patterns develop where these abstract vector like empty entities can merge or separate. Two lovers may break up and still live together for all the outside world as if defined in a Platonic memory past the state of drastic change of relationships if in the world the imbalance only to be compensated by the interpretation of patterns of their dreams.
Such a state my work together both ways...given some formal way on a four way world, quasifinite in that with time itself we may not strictly define what is inside or outside, just as the binary helices that merge or separate in biology, that is time needs not be absolutely one directional outside or inside some past or future or middle zero stance of a state of experiencing and learning mind, in complex space many issues of the grounding of theory is what are the space like or time like dimensions. In complex space the abstract count of so many seemingly isolated singularities (holes so to speak) these may make knots possible in matters of rotations and so on just as they are only possible in real three space as stings or lines. We double such structures and double them many times as these are abstract doublings but have to be justified in the formalism not by the simple quantum directionality with its own uncertainty and coherence but the ensemble of global logic that contains yet evolves the unaware subconscious machine brain as well an adult intellectual assessment, so too the teleology as far at it goes as our intuition.
While there appears something deeper than quantum theory the parallel or many-world view as quantum computers adds little to the understanding or manufacturing of the whole, say not as something relevant to military use, but quasic ideas do seem, sometimes close to the middle ground of triviality, to justify the classical and quantum mechanisms where they work as physicality. We do not find, as a matter of probability perhaps, what we are not looking for... and we do not conjure from the nothingness what we do not have the threshold of awareness to imagine. But the ideas of Godel and Turing in the ensemble of machines suggest something more than a last Omega involves, thus choices beyond that at the frontier of natural law yet universal freedom as structural systems as universal law to contain, abstractly, the one and the many, the biochemical like pathways of delay that forms a utilization of processes linearly or in mazes of time.
We should not make such machines save for the experiments sake in times of sufficient funding knowing such principles nor expect results from complexity of grouped machines alone constrained by Omega universal machines or ever receding last absolute ideal infinity as with Cantor's Omega. We may only reach some sort of teleological Omega in the sense of time as such a direction of progress, of intuitive time with this question of centers, beginnings or ends. But as things are in our time, if we are concerned with vulnerability between states to which disputes sometimes in the balance at the fulcrum of the seesaw reach the states where surprises come outside a system, such basic research if neglected or suppressed in a society amounts to the new dark ages in the main, dark in the reflections of our minds and mindless behaviors. To such an unknown frontier even as we have the ability to judge scenarios, to abandon this sort of project and fundamental research makes us more vulnerable than we have ever been.
We are vulnerable also to close minded self assured ideas where they cannot distinguish right or wrong so decides if an artificial machine is immoral as to its maintenance or saving as it not being human but being subconsciously so in its architecture that does not need clockwork time to evolve, for can the mentally ill be so on this wise if in a sense they have no mind? Yet let us not hold sane minds away from free inquiry and weapons, to disarm them is to make the general state of our society less than sane, to declaw the cat as acceptable so they do not scratch the furniture.
On the other hand, that we, even an individual exists so as to awaken to an new concept it may set off parallels in the social whole for these paradoxes and debates continue the interpretations as to how consciousness engages with the universe and its deeper realities. Communication and signals are but parts of the divinity of intuitive patterns.
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Footnote: considering the mapping of functions, and that certain things we cannot follow due to difficulty between odd and even numbers (I mean sometime or other we might ask if zero is even for example) but like the sine we have functions that lead (in a simple oscillating way) to its negative. I must think more on this as that distinction is not clear in my algebra...the numbers have gender idea. But do not we have a core issue with particle and wave as conceptions? Something in this idea suggests to me a key issue at deeper foundations- in any case it may relate to this post as to how we treat the simple division of these abstract and countable creative hole objects- still the concept of a hole may in a sense be a non-existent idea unless we mean in a positive (reduced to zero) sense it counts genus as holes all the way thru. Stashef once agreed with my comment that there were no such things as holes back when our local stringers consulted him on topology (interestingly he also was drawn to work on quantum bits.)
Some sort of structure (stereonometry) goes beyond the metaphysics of physics foundations. Far beyond such a use of these for quantum computation speculation. Even there forces and particles may come in nonnecessary innate pairs. Either such negation is a matter of convenience that undermines necessary redundancy or it is a sort of physical information itself that does compare intervals irrational and transcendental- perhaps the first frontier to find new mathematical physics beyond.
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