some thoughts from the Pensingers
These many years after the General Process paper, we are getting better insight into how the multivalued reference space has to be constructed on Hilbert space under m-valued logics. Musculpt and the hierarchy of c-s, c'-s, and c''-s are right at the heart of it. If these insights are pulled out of their m-logically-valued context and put into 2-valued syllogistic logical march via written notation, they will have lost all their intrinsic meaning. On the contrary, cognition has to be pulled out of march in logical syllogism and let fall into Musculpt. Absent Musculpt as mathematical notation, circular presentation is the only real approach, because engagement with it forces the visualization pre-requisite to conscious emergence of always-there subliminal Musculpt (which conventions of written notation deny).
Our tornado-genesis-related speculation that interval spread in the hierarchy of c-s, c'-s, and c''-s is in natural log distribution clearly relates to the N/logN distribution of primes, made more precise by Euler's zeta function relative to only the real numbers. Riemann generalized Euler's function to the imaginary numbers and identified a critical band within which all the primes must fall. That they all fall on a line within this band is the famous Riemann Hypothesis. In order to encompass the hierarchy of c-s, c'-s, and c''-s, the Riemann zeta function will have to be generalized to hypercomplex numbers (à la Charles Musés) and the critical band will have to be mapped on multiple sheets relative to m-valued logics. (Distribution of c-s represented by Euler's zeta function relative to the real numbers; distribution of c'-s represented by Riemann's zeta function relative to complex numbers; distribution of c''-s represented by a zeta function relative to hypercomplex numbers.) In this manner, issue of trivial versus non-trivial solutions (sets of zeros) will disappear.
What will happen is this: on each prime (arrayed on Riemann's line) on the m-logically-valued reference sheet will be stacked other primes from the multiplicity of single-logically-valued sheets composing the Riemann surface map of Everett’s universal wave-function. (In this approach, Cantorian fractal spacetime relates to the stack of single-valued sheets, which, in turn, relate to Sakharov’s collapse/anti-collapse multi-sheet model of the universe.) On each of the multiplicity of decomposed single-valued sheets, Riemann’s line will be located differently within the critical band than it is located on the m-logically-valued reference sheet, such that, when the complete superposition of numbered sheets is considered, the line will have spread across the whole critical band on the reference sheet (as a result of expanding consideration from single-valued logic to logics of m-values). Because the hypercomplex zeta function would represent distribution of limiting velocities, accelerations, and time rates of change of acceleration, the waveform configured by the distribution step function would be an idealized chronotopological invariant characteristic of the genus (connectivity) of that universal covering surface which is the reference state of a perfectly efficient autopoietic process in optimum self-correlation (which is anything but a catastrophe! to all those not identified with the ego-complex).
A supradense m-logically-valued Hilbertian reference space constructed in this fashion has nonlocality of embedded objects as a fundamental property. Locality is a decomposition issue involving cycles of self-reentry (or, alternatively stated, of cosmological self-forgetting,
amnesis -- while recomposing the m-logically-valued reference space is Plato’s anamnesis). Lesser levels in efficiency of autopoiesis have chronotopological invariants based on proper subsets of the primes, each with their characteristic step functions and waveforms. (Though these subsets can be put into one-to-one correspondence with the set of all primes, their distribution patterns vary.)
Gödel used prime numbers to encode each element of a logical statement. He defined a specific prime number as corresponding to the equals sign, for instance. And another for plus, and another for minus, and so on. The product obtained by multiplying together all such prime number factors (which, taken together, encode the given logical proposition) is called a Gödel number. So, all possible statements have their equivalent Gödel numbers. Any real number can be factored uniquely into a product of prime numbers.
This Gödel methodology for proving his famous theorem is, of course, a very sophisticated form of gematria, sharing properties with Kabbalah -- the portion of gematria missing from Gödel's treatment being the point-set topology associated in Kabbalistic thought with each Hebrew, Greek, or Sanskrit letter. In the present context, consider Stan Tenen’s discoveries, and consider them relative to operator-time. (Operator-time would be characterized by the hypercomplex zeta function, which we believe our canonical equation for harmonic temperature oscillation of
p-electron parcels of superconductant DNA can be understood as - via Julia’s work on primes and critical Hagedorn temperatures.) Tenen’s topologically configured strips (gematric equivalents to the first sentence of Genesis in Hebrew) are Gödel-encoded segments of Riemann’s critical band (as it meanders the multi-sheeted Riemann surface) corresponding to invariants of classes of temporal operation on the m-logically-valued reference space.
So, were the Gödel-numbered point-set topologies of Gödel-encoded m-logically-valued propositions to be Riemann surface mapped, every point on the multiple single-valued sheets would have a number associated with it -- some prime, some non-prime. Logical propositions of m-1 and less values would be equivalent to lattices of prime-number points, the values of which are multiplied together (such multiplication being very much like neuronal firing pattern sequences mapped on Szentagothai’s multi-sheeted model of the cerebral cortex). The Gödel numbers resulting from this multiplication would be located on points on the m-logically-valued reference space (the most densely point-packed Riemann surface sheet). The fully-Gödel-numbered m-logically-valued reference space would, thus, contain all possible propositions (which is a very good notion of a degenerative universal grammar, i.e., based on archetypal decomposition, rather than recursive generation) in superposition of factorial involutes mapped as lattices on the decomposed single-valued multiple Riemann surface sheets. We believe these lattices are Regge lattices, and the involved propositional calculi, Wheeler’s pregeometry.
Gödel did not prove that arithmetic is more fundamental than logic; he proved that arithmetic is more fundamental than 2-valued logic. Gödel did not prove impossible Leibniz’s dream of a universal calculable language; he proved such language (i.e., Musculpt, laser Esperanto, holographic Volopuk, logovisual technology) impossible under 2-valued logic. Just as written mathematical notation is inadequate to transcribe m-logically-valued modes of thought (which require Musculpt as mathematical notation), so proof is no longer an interesting mathematical problem/exercise: no obvious self-evident axioms; no final or first cause; no ultimate decidability, making all decidabilities relative.
Proof is a relatively meaningful, but fundamentally meaningless, move in a glass-bead game. Other mathematical activities are far more interesting: injunctions, interrogatories, proclamations, constructions: in the real world, a priori and a posteriori are identity-transparent Kantian categories of unus mundus. The physical universe we are conscious of is the unconscious we strive to bring into consciousness. Not in(t)here is not out(t)here! Non-orientable self-reference is a matter of difficulty only under 2-valued logic. Not only is the thing-in-itself unknowable; not only is there no thing-in-itself to know; but the itself purported to be distinguishable from the thing to be known is no-thing, knowable or unknowable.
We believe that the rules of dodecaphonic music composition Schoenberg evolved in the second decade of the 20th century (in Vienna with Kandinsky, where the two briefly exchanged the roles of musician and painter) for post-atonal composition can be applied to Gödel encoding of logical propositions, such that the universal grammar, contained in the transfinite set of Gödel numbers on the m-logically-valued reference space, can be displayed/played as Musculpt. We believe Kandinsky's ideas, like a triangle can only be yellow, explicated in On the Spiritual in Art (1910), was in direct reaction to Planck’s notion of a quantum of action and an attempt to crystallize the synaesthetic colored-hearing aspects of Musculpt he was soon to paint as universal grammar of form in process, which can be rigorously developed by studying frequency correlates of Gödel numbers which involute into logical propositions arrayed on the multiple sheets as lattices. The harmonic statements plotted on dodecaphonic composition matrices can be stacked, and clearly can be Gödel encoded, which means that an m-logically-valued argument can be played back musically and holo-sculpturally. Or, vice versa, music-sculpture topologies can be decrypted into their equivalent logical propositions.
Gödel’s definitions on primes, relative to equals, plus, minus, and so on, thus, need to be empirically verified relative to nature’s form in process. His specific choices for prime number codons were made on basis of the glass-bead game play which concerned him. Other choosing algorithms related to classes of natural process (such as severe storm genesis, DNA generated coherent waves) would establish other sets of prime number codons generating other classes of Gödel numbers (including those generated under m-valued logics and axiooooomitized; relative to non-self-identical numbers in MOON).
But this direction of formulation is ego-complex inverted from the unus mundus case. The universal consciousness (collective unconscious from perspective of the ego-complex) moves as Musculpt. In order for Musculpt movement to enter (more accurately congeal) a brain, trigger neuronal firing patterns, and thus engage in spontaneous localization, it must involutionally Gödel encode: this is why the ancient world was so transfixed by gematria, by stellar configurations (point sets) as astrological propositions, injunctions, interrogatories, proclamations, constructions. Paradoxically, and tragically, the Inca appear to have succumbed to 200 Spaniards by being thus transfixed. The question is, Is there a universal grammar of Gödel encoding for the hierarchy of Russellian types of Gödel numbers?
What is most interesting to me about Gödel's work is that he chose to ignore Emil Post's m-valued logics, which were on the scene for over a decade when Gödel published his famous proof. This choice on the part of Gödel is roughly on a par with the performance of G. Birkhoff and J. von Neumann at about the same time, i.e., three years prior to the Nazi invasion of Poland, the home ground of m-valued logics. (See: The Logic of Quantum Mechanics, Annals of Mathematics, 37, 1936.)
One truly must wonder at the extraordinary lengths the mind is willing to go in order to avoid looking the multivalue straight in the face. Here, Birkhoff and von Neumann recognized that quantum logic has some relation to projective geometry, but where is the Riemann surface stack? Though some doubt is cast on the utility of Hilbert space, where is the recognition that every point in the referencing phase-space is multivalued, that translation across a single-valued sheet is projected as a static lattice to the multivalued referencing function-space? How strange that they should invoke the concept of a logical lattice, but not view it as a true point-set topology!
These logical lattices of Birkhoff and von Neumann, of course, are of the essence of Post's mTm
logics (1921). Standard Aristotelean-Baconian-Boolean logic is a 1T2 logic. Two values are available, one of which is permitted per proposition. This is logic reduced to truth-value and all the standard syllogistic rules. Post generalized this to the mTm case, where all logical values available are permitted per proposition.
On the Riemann surface stack of single-valued sheets, there are 1, 2, 3
values available on successive sheets, but only one permitted per proposition (represented by a given logical lattice of connected points). All the values on equivalent points on each such sheet are stacked on their equivalent point on the multivalued reference space (most-dense sheet), where there are m-values available and m-values permitted per proposition. Gödel numbering of these m-valued logical lattices necessarily involves expanding the universe of discourse which has grown up around the famed Riemann Hypothesis on distribution of prime numbers (which are factors of Gödel numbers).
With similarly inexplicable myopia, Birkhoff and von Neumann used involutory relations and the concept of skew-fields, but the multivalue screaming in the background, without which skew-fields are impossible, was completely ignored (along with the whole notion of skew-parallelism, qua skew-perpendicularity, underlying skew-fields, with its profound implications for Riemann's thesis on the origins of charge, Heisenberg's indeterminacy, and resultant non-conservation of energy and virtual phonon exchange underlying mechanisms of high temperature superconductivity, such as that of DNA). Dropping distributive laws for 2-valued propositions, indeed! Can laughter be suppressed? B and vN knew of Post's logics, just as did G. These missing recognitions by Birkhoff and von Neumann are not ignorance speaking; they are expressions of psychological dread. And this paper, summarizing at the very least a decade of collective psychoneurotic posturing, was published a mere three years before the inevitable avalanche of consequences!
In arriving at his famous proof, Gödel falsely assumed that: (1) language A is orientable, i.e., that in language A can be absolutely distinguished from not-in language A; (2) infinite regress is linear in nature, i.e., that regresses are not self-reentrant marches; (3) description reduces to truth tables, i.e., that there is nothing in logic more fundamental than truth-value; (4) logical calculi are limited to only one order of logical-value, i.e., the 2-valued variety; (5) single-valued propositions are the only legitimate propositions, i.e., that self-contradiction is a violation of the rules of all logics; (6) completeness is an absolutely meaningful notion, i.e., that there are no languages for which completeness is undefinable -- which is not the case for languages involving use of logical calculi with orders of logical value greater than two and propositions with logical values greater than one.
We wonder, What is the Gödel number of this 6-fold proposition?
Gödel numbers need to be plotted, not calculated! More accurately stated: factors of m-valued Gödelized propositions need to be plotted on the Riemann surface sheet stack composing the Musculpt manifold, not calculated to supposed single definite values which non-self-identical numbers do not authentically possess. The collapsed-value of a superposed number (such as a Gödel number), localized on the Cartesian grid, is not the full measure of this number. Single-valued Gödelized propositions are a special case (likely a trivial case, cosmologically speaking) of all the m-valued Gödelized propositions there are in this world.
Non-self-identical superposed numbers are not numerical values! They are FORMS of Musculpt, i.e., archetypes. Arithmetic operations involving such numbers are carried out by operator-time. Non-self-identical superposed numbers are dreamtime configurations of songlines (like Riemann's critical line). How are such arithmetic operations represented?
Factors of m-valued Gödelized propositions are plotable, even if their multiplied values are Turing uncomputable. The resultant plotted lattices are Regge lattices, which are transformable into skew-field curvature configurations by the Regge calculus -- which transforms Einstein's field equations into n-dimensional lattices, and vice versa.
Where do these lattices underlying the semantic sounded-forms of Musculpt fall? Not on the real line. Not on the complex plane. The s (a complex number) of the Euler and Riemann zeta functions is logically 2-valued. Its 1/2 + it does not conform to the requirements of mTm as t ranges. The zeta function's s, like the nk landscape of complexity theory, is formulated in terms of ordinary binary logic. The nk landscape is actually a 2nk landscape (for a fuller account of this see: M-Valuation in a Generalized Currency Basket) where the elements in question must be either on or off.
This is a highly relevant comparison because k indicates the number of correlations between the involved elements or factors, n, and, also, because non-self-identical numbers under m-valued logics are numbers in superposition (i.e., correlated) stacked on a point in the reference space. Generalizing the landscape under mTm yields an Mnk landscape.
Were the elements or factors arrayed on this landscape prime factors of Gödel numbers of m-valued propositions, the s of the zeta function would become sM. Now, the power set (set of all subsets of the original set) in Cantor's continuum hypothesis has 2n members, just like the nk landscape of complexity theory has 2n elements or factors. This is a statement about cardinality: the power set has cardinality n.
The power set of non-self-identical numbers of cardinality n, however, is 2nk -- correlations being involved in the non-self-identical. But this is an incomplete statement, because we are not talking about only the relationships between members of the set (i.e., the many ways in which they can be arrayed into subsets), but about their changing animistic identity transparency under increasing orders of logical-value, m. The power set under m-valued logics is not identical to the power set under 2-valued logic.
The m-logically-valued power set of non-self-identical numbers of cardinality n, is actually Mnk, where M is the order of logical-value and k the number of correlation factors (of the involved logical lattice with n bifurcation points). Under such considerations, however, Riemann's zeta s could not be merely a complex number; it would have to be a hypercomplex number, because the complex plane is a 1T2 logical construct.
Constructs of order mTm are beyond the complex plane. sM is the order of hypercomplexity of s. The m-logically-valued power set of non-self-identical numbers, sM, of cardinality n, is . On this power-set continuum, not Cantor's, can we begin to plot the Gödel number factors of the m-valued logical lattices underlying the sounded-forms of Musculpt semantics.
It is only non-self-identical numbers that make the notion of cardinality of the continuum plausible. Degrees of identity transparency, of order m, between the discrete and the continuous, dictated by mTm orders of logic, is the fundamental modification of Peano's axioms required. It can be seen that under Post's logics there is no limit to the number of infinities between that of the natural numbers and that of the continuum, such that, for instance, the critical line of Riemann's hypothesis has a unique shadow line under each order of logical-value greater than 1T2.
Actually, we do not regard consistency an inherent attribute of any possible world semantic, except those world semantics of the 1T2 variety. But, then, I do not regard any possible world semantic as merely possible, for the possible 1T2-variety world semantics are actually treatments of the variety of mTm world semantics as if they were merely possible -- merely possible, that is, because consistency is mistakenly considered an inalienable right of world semantics qua world semantics. I must, however, protest that this is no small, purely theoretical, matter in an esoteric corner of one of the myriad fields of logic. If, as regards the 1/2 + it of the Riemann zeta function, the t º a Hermitian operator on Hilbert space, as some have suggested (including Hilbert himself), the semantic (import of the involved quantum logic, that is) of some possible (in consensus quantum theory, probable, however improbable) world is subject to interpretation under mTm such that it becomes iTM, where T would be an m-logically-valued Hermitian operator on an m-logically-sheeted Hilbert space. In such a Hilbert space under such a Hermitian operator, the idea of consistency one can entertain becomes a very peculiar notion, indeed. The orders of truth table multiply without bound. Moreover, the double-exponentials associated with such truth-tables indicate they literally turn an imaginary corner (truth table deposed in hyperdimensions) in propositional space, meaning that the i of iTM indicates, not only that the involved numbers have an imaginary component as Riemann's zeta function indicates, but, when given any interpretation whatsoever, so must their resultant contextual meanings have imaginary components: when 1T2 goes to 2Tm to 3Tm to 4Tm and so on, the imaginary semantic dimensions of the involved worlds multiply hand over foot, as consideration of G. Spencer Brown's calculus of indications would indicate.
In face of unbounded and exponentiated orders of truth table, we personally prefer superposed actual, as opposed to separate possible, world semantics. Just as Everett-Wheeler-Graham (EWG) argued on basis of linear-time-bound thermodynamic considerations that their multiple universes (mapped in a 1T2-logic-constructed Hilbert space) are incommunicado, in solitary confinement, so Saul Kripke, in constructing world semantics on basis of possibilities, rather than actualities, treats, by direct implication, any meaning-intersections between worlds-of-meaning as tantamount to trivial: worlds of horse-sense, pig-sense, dirty-rat-sense, and so on are overwhelmingly 1T2 worlds unto themselves. Both the EWG and Kripke cases so transparently justify the individualistic mode of ego-sphere-identity, caught in inherent alienation and anomie of Existential separatism, one is inclined, on that basis alone, to reject them.
Saul Kripke has long been committed to this notion of identity as the only possible world semantic of identity (as a Kantian category). In 1976, if memory serves, Kripke delivered a lecture at Cornell University mistitled in campus promotion as Time and Eternity. He began the public lecture by informing his audience that the correct title was Time and Identity. Everyone had a good laugh, as all knew no reputable philosopher would be caught these days lecturing on eternity. Some of the more informed in the audience, however, would have been the more happy had the logician lectured on eternity as opposed to identity in relation to linear-time, or more happily yet on eternity in relation to identity, as the monotonic notion of identity he derived (implicitly as a Kantian category) from linear-time-only so thoroughly neglected the variety of identity constructs experienced historically by the human species, one had to wonder whether or not the promotional snafu was a possible cosmic joke. This possible joke, if it was indeed actual and cosmic, was all the more poignant in that the imaginary corner inevitably turned in propositional truth-table space through double-exponentials is an imaginary turn away from truth-value toward animistic identity transparency, which imaginary turn Kripke's derivation of monotonic identity from linear-time-only implicitly rejected -- which rejection, of course, is a foundational assumption of his "possible world semantics", i.e., configurations of meaning arrived at through nonmonotonic reasoning about monotonic identity.
William L. Pensinger & (Cong Huyen Ton Nu) Nha Trang Pensinger
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