Deep Mathematics and Meta-systems Theory by Kent D. Palmer

Metaphysics and Mathematics Deep Mathematics and Emergent Meta-Systems First we define the metaphysical in the Theory terms used by Anaxamander as the distinction between the Apeiron

Kent D. Palmer (Unlimited) and Peiron (Limited). There [email protected] are many metaphysical principles in http://kdp.me Western Philosophical history, but the 714-633-9508 Apeiron was the first of these. The most Orange CA 92867 common interpretation of the

Copyright 1997, 2014. All rights reserved. metaphysical principle is Being. The Finished 971219 - Not for Distribution. peiron is distinguished again between Corrections: Draft 0.8 2002.12.10 deepmth8.doc the Logos (upwelling of speech) and Physus (growth of physical things). Once these two dualities are defined Introduction there are myriad interpretations of the different terms involved and their In this paper we will explore deep relations to derivative terms. However, mathematics and connect that with the we will take the position that all these theory of meta-systems. Deep interpretations form a field. The field is mathematics is a new extension of the such that each distinction like a mobius domain of mathematics through the strip is locally dual and globally non- consideration of ontological matters that dual. But around the mobius strip every are foundational for Systems Theory. possible demarcation is at one time or We extend the Systems Theory or another designated as real. In order to George Klir in Architecture of Systems avoid this situation we posit that it is Problem Solving with a Meta-systems possible to prevent the logos from Theory. We then place both Systems interacting with the physus thus Theory and Meta-Systems extensions in preventing dualism from taking hold. In a broader mathematical framework. such a case we recognize nomos as the Most of this deepening of mathematics non-dual order that is archaic prior to the follows from the question concerning logos/physus distinction. Similarly we what is more fundamental than can disarm the dualism between the mathematical Category Theory. We Unlimited and the Limited by forcing attempt to develop this foundation in the them apart and thus recognize the context of the development of a theory fundamental Indo-European non-dual of Emergent Meta-Systems. Emergent moment prior to the arising of the Meta-Systems stem from the recognition Apeiron/Peiron distinction called RTA. of three special systems that act as a RTA means cosmic harmony in the hinge between Systems and Meta- Indo-European tradition and stands for systems. The attempt to understand this the source of order. RTA is translated in hinge drives us to understand the our modern English language as Right possibility of non-duality and attempt to which designates the preferable frame a mathematics that can approach asymmetry and that which is the comprehension of the non-dual. contextually correct based on principle.

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Given this metaphysical landscape we Heidegger’s ontology that solves the can immediately recognize the nomos as problems of Kantian metaphysics the object of Mathematics as a without giving up its critical boundaries. discipline. Nomos is ordering which lies Just as a fourth dimensional space allows behind both the unfolding of the Logos us to exit a sphere without going through and the Physus. It is because the Nomos its boundary so the Process Being is prior that physical states of affairs can modality allow us to exit the realm of be connected by mathematics to our critical transcendental philosophy theories that unfold in the logos of without violating its precepts that keep scientific discourse. If the nomos were Reason engaged with understanding. not prior to the arising of both aspects of Heidegger's monolithic Being with its our duality at the epistemic level this two modalities appears as an archaic connection would not be possible. The origin prior to the arising of the non-dual RTA is the principle that lies Subject/Object (logos/physus) behind that non-dual order as its source. dichotomy. In this way it attempts to RTA operates in the chiasm between the approach the non-dual by positing that Unlimited and the Limited while the the Apeiron is the Monolith of Being nomos operates in the chiasm between with its dual modes and that the Peiron Logos and Physus. Chiasm was a term arises from that giving us the introduced by Merleau-Ponty to describe subject/object split which transcendental a state of Being very close to non- Critical Philosophy depends upon. duality. Apeiron/Peiron and Heidegger discovered a ground which Logos/Physus form a mobile. RTA and combined the views of the Presocratics the Nomos arise within the interspace of Parmenides (Pure Presence) and that duality when it is disabled. As long Heraclitus (Process Being) in a neat as it is operating dualistically these non- formulation. Heidegger used this to dual moments are invisible but as soon propose a solution to the great split that as we put the duality out of play then the had appeared in physics in his day which non-dual moments appear. was between Quantum Mechanics and Relativity Theory. Both posit a globally As mentioned previously Being is the incoherent non-experience-able state of normal interpretation of the Apeiron. In affairs contrary to the coherent state of our time Being has become fragmented. affairs what we perceive. In Quantum What was once the single highest Mechanics that is the probability wave Concept has now become internally which is simultaneously in all possible differentiated in a series of steps. The states that collapses to one state original concept of Being is called by randomly when perceived. In Special Heidegger in Being and Time a Pure Relativity Theory that non-experience- Presence Being (present-at-hand) which able realm is four dimensional he distinguishes from a Process Being spacetime. We experience this (ready-to-hand) as two modalities of background matrix of all phenomena Being proper. Being proper is a monolith differently from different inertial frames which embodies transcendence of reference. In this case the Lorenz grounding itself. These two modalities transformations between different operate within the monolith of Being as observers is not relative at all but a kind of higher dimensionality within mathematically precise. Thus Special

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Relativity Theory is inherently deterministic because it is rooted in Pure deterministic while Quantum Mechanics Presence. Being itself is split into two is inherently probabilistic. But these two modalities in a complementary fashion. incompatible views of nature are duals The two major views of physical theory of each other, and can be seen as sharing that extend the Newtonian paradigm the positing of a non-experiential realm arise from the duality inherent in the like the monolith of Temporal Being that monolith of Being. includes the two modalities of Pure Presence and Process Being. Our All would be well if there were only two subject/object experiences arise out of modalities of Being within the Monolith. this more archaic level of Being. Our But continental philosophy discovered experiences are objectively three that the Monolith breaks up by the dimensional slices of a four dimensional emergence of two more levels of even spacetime matrix. Our experiences are more radical difference that that subjectively observations that collapse suggested by the term mode. Rather the probability waves that inhabit the quickly it was recognized that Process four dimensional realm. In either case it Being had an antinomy called is our experience that brings the Nothingness by Sartre in Being and phenomena out of the Monolith of modal Nothingness. Although, Heidegger Being into one of its modes. One of attempted to dismiss Sartre's less those modes is like the nouns of sophisticated brand of Existentialism, it language while the other is like the became clear that the antinomy was very verbs. In other words we either have an significant and in fact broke the emphasis on operands or operations. But Monolith of Being into pieces. Thus, the Monolithic Being itself is unified in with the advent of what Merleau-Ponty the sense that it is both an operator and called the Hyper Dialectic between an operand at the same time. This, in Process Being and Nothingness in The fact, is the Indo-European ideal of Visible and the Invisible, and what wholeness. This fact that Heidegger's Heidegger called Being (crossed out) a philosophy was designed to solve the new kind of Being appeared and quandary of the duality between converted the previous modalities into quantum mechanics and special kinds as well. Derrida capitalized on relativity is a little known facet of his this work and defined what he called philosophy. Heidegger, studied physics differAnce as the supplement of after Theology and before Philosophy differing and deferring which he and had hoped to become a Physics explored in terms of the fundamental Professor. Philosophy was his third difference in the Western tradition choice. His philosophically sophisticated between the place of Speech verses the and subtle solution says that Special place of Writing. I call this kind of Relativity Theory and Quantum Being -- Hyper Being and assign it the Mechanics are duals of each other like modality 'In-hand' because it is the point the particle/wave or the measurement of where radical transmutation takes place position or velocity. They both arise and the technology that appears in the from the same source. One is ready-to-hand of Process Being probabilistic because it is rooted in transform in our hands. The being-in- Process Being and the other is the-world, i.e. Dasein (being there)

3 Deep Mathematics and Meta-systems Theory by Kent D. Palmer which is the human prior to the arising actually goes beyond Being into the of subject/object duality has different realm of existence and is actually non- modalities. For Pure Presence the dual. We interpret this existence of the modality of being-in-the-world is called fifth meta-level of Being as over the present-at-hand by Heidegger which edge and outside Being proper in terms Merleau-Ponty explains psychologically of the Buddhist concept of sunyata in Phenomenology of Perception as (emptiness). This makes our ontology “pointing.” For Process being the empirical in the sense that anyone can modality of being-in-the-world is called try to 'think' the fifth meta-level of Being ready-to-hand by Heidegger and and thus expand our world. But as long interpreted psychologically by Merleau- as this remains a challenge unmet then Ponty as “grasping.” I augment this by we say that what is at the fifth meta-level giving Hyper Being the modality of and above is outside Being. Being is not being-in-the-world as the 'in-hand' and 'n' dimensional but only has four meta- giving the psychological interpretation level emergent steps. After that the stairs taken from Levinas as 'bearing'. Levinas stop abruptly and whatever is beyond says that in this third kind of Being, that is outside our worldview. We give Hyper Being, is the point where the name Existence to what lies beyond Metaphysics and Ethics collapse the pale of the highest threshold of together. Being.

But the radical difference of kinds that Being proper has four mode/kind/integra arises between the modes of Being at the differentiated moments. It also has four third meta-level of Being is not the end aspects related to reality, identity, truth, of the emergent series. At the fourth and presence. We define existence as meta-level a new even more radical that which is neither true nor false, real difference I call the Integra arises. The nor unreal, identical nor different, integra is an ordering that goes deeper present nor absent. Being proper is the than essences and orders the individual process of presentation and ideation. at fractal levels within the accidents that That is Being proper produces illusions. make up the individual. This produces a What goes beyond Being and has only new integral unfolding of Being which existence is what we find (Arabic wujud) Merleau-Ponty calls Wild Being in The without presentation (Arabic kun) mixed Visible and the Invisible. I give this final in. When the Arabs interpreted the sort of Being a modality of being-in-the- Aristotle they noticed that Being did not world called "out-of-hand" and say that exist in their language. Thus they its psychological interpretation is invented the technical term Kun (made "encompassing". Wild Being is almost thing) to stand for Being in order to impossible to think and all the higher differentiate it from Wujud (existence) meta-levels of Being are actually found in Arabic naturally. Then, when impossible to think, similar to the this scholarship was taken back into the unthinkability of Bateson’s fifth meta- Latin the word Existence was coined to levels of learning and motion in physics stand for Wujud in contradistinction to as described in Steps To the Ecology of the essences that had Being. There has the Mind. Wild Being is almost non- been an implicit tension between essence dual while the fifth meta-level of Being and existence ever since which came to a

4 Deep Mathematics and Meta-systems Theory by Kent D. Palmer head when Jaspers made an issue of it. form there lies three other levels in the Then the existentialists followed Nomos that are unexplored hither to in emphasizing existence over essence as the Western Mathematical tradition. It is Sartre did. Heidegger accommodated the the purpose of this paper to lay out these two and interpreted existence as the other ontological levels of possible production of subjectivity/objectivity out mathematical elaboration as we move of the self-grounding monolith of Being. from pure duality and pure continuity Thus Heidegger interpreted existence in toward the non-dual and chaotic terms of ex-stasis or ecstasy of the discontinuity. Here we have identified a projection of reified beings by our being- new frontier. We will survey that new in-the-world out of the Modal Monolith, landscape but it is so vast that it will the It of It Gives or what Heidegger have to be left to others to discover what latter called Appropriation in contra other wonders exist there. distinction to the enframing of the experiential Newtonian realm. Unfolding the Nomos We have identified that all mathematical Pure Process Hyper Being Wild Being Presence Being categories hither to identified and No Modal Kinds or Integral explored exist only in one ontological difference difference essential difference difference layer. We will trace a route to the Present-at- Ready-to- In-hand Out-of-hand deepening of Mathematics starting with hand hand category theory. That route will take us Point Grasp Bear Encompass Determinate Probability Possibility Propensity away from our idealized Eldorado into a Form Pattern Trace Disposition/ jungle that is ignored by mathematical Tendency formalism and logic. Or at most Now that we have defined the integral mathematics constructs images of these kinds of Being and shown how the other levels within the realm of pure undifferentiated and continuous concept presence. Mathematics never leaves its of Being that existed up to the present haunts where in formalisms can be easily century has become fragmented in built. These other ontological levels are modern Continental ontology we can successively harder to think about less frame our major mathematical point. All well build mathematical representations mathematics is basically Present-at- in so as to explore the nomos. As we hand. Mathematics has not made the move through our exposition of the fundamental transformation that both unfolding of the nomos we will contrast Philosophy and Physics have made into that with the unfolding of the physus and comprehending phenomena thorough the logos. In the realm of the physus we different ontological modalities. Every have formal structural systems theory mathematical formalism is setup as which we apply as a methodology and in purely present and operated solely the realm of the logos we have logic. To within that modality as if that were the set the stage we will describe the only ontological modality that existed. segmentation of our worldview. This historical fact makes Mathematics with all its differentiation merely a In the Western worldview there is as we surface phenomena considering the have said the realm in which we make unexplored depths that lay beyond the distinctions. These distinctions are formal surface. Beneath the surface of designated as real, or identical, or true,

5 Deep Mathematics and Meta-systems Theory by Kent D. Palmer or present. Against this backdrop there is templates for understanding that may be the continual pressure of analysis against applied to any level of the ontic the physus by the continual use of hierarchy. The ontic hierarchy is reductive methods. Theories on the other produced by the failure to be able to hand have to withstand the continual analyze and reduce everything. What pressure of skepticism. We apply this ever withstands extreme attempts at reduction in the following manner. We reduction is generally accepted as first identify a form and attempt to emergent ontic levels in the physus. construct a formalism within which What keeps the Emergent Ontological proofs are possible to understand it. If levels independent of each other is a we cannot understand the form through a healthy skepticism that every theory that formalism then we reduce that form to unfolds from the logos must face. The its contents and attempt to use a category ontological emergent levels are merely scheme of contents to explain it away as common strategies of understanding at a means of understanding. If we cannot various levels of complexity. All these reduce it structurally then we fall back levels are advanced tentatively in order on description of the wholes that the to give us something to talk about but form is part of using systems theory as are not in any sense final or an underpinning. Formal Proof is the demonstrated to have any kind of strongest strategy of understanding, identity, truth, reality or presence in Structural Explanation is the second themselves. They are merely designated strongest strategy, and finally Systemic true, present, real, or identical for the Description is the weakest strategy for purposes at hand. attempted understanding. Basically if we cannot prove we fragment in order to The strategies for understanding and a explain and if we cannot explain then we healthy skepticism concerning their accept wholeness. This fall back to application is what gives us systems wholes as a last resort causes us to posit theory and its extension up and down the emergent ontic levels in the phenomena ontological layers. These appear as like: models in the theoretical logos. The Ontic (physus) Ontological (logos) inner coherence of these is maintained in (Gaia) Pluriverse terms of Logic. We contrast this to what Social Kosmos becomes present phenomenologically Organism World which is tested to ascertain its reality on Multi-Cell Domain Cell Meta-System the side of physus. The move from what Macro-Molecule System appears as the thresholds of complexity Molecule Form of theory toward logic on the one hand is Atom Pattern the dual of the move from what is Fundamental Particle Monad presented through testing toward the Quark Facet reality of the ontic emergent levels that (Sub-quark) inform the physus. So the hierarchy of models is contrast to the logic that gives There is no inherent relation between the it coherence on the one hand. The two sides of this dualism. We get them hierarchy of emergent phenomenal mixed up all the time saying that levels is contrast by reductive testing to Systems and Forms exist in nature. But what appears. Between these two realms fundamentally the Ontological are

6 Deep Mathematics and Meta-systems Theory by Kent D. Palmer of physus and logos appears the non- onto the associative property. Thus if we dual nomos that informs each. We get at want to produce a move toward the the nomos by using mathematics as a indeterminate away from the determinate means of rigorously underpinning our present-at-hand basis of current models. We use experimentation to mathematics then we will have to give verify that these models actually capture up the assumption of the associative what is happening in reality behind the property. This will produce something appearances. Ultimately the Kantian will similar to a figure found in logic called tell us that the transcendental object is a the Square of Contraries and noumena that cannot be apprehended Contradictions that is used by Greimas either by reason or experience. Also the for a structural analysis of narrative. We Kantian will tell us that the follow Greimas and note that this transcendental subject is similarly structural analysis based on a form from inaccessible. It is the transcendental God logic is precisely what we need to move that maintains the coherence between away from formalism to the level of these two syntheses in the world behind pattern. Grenander in his books on the scenes. But Heidegger tells us that Pattern Synthesis is the only exactly that thing which Kant took out of mathematician we know to attempt to his metaphysics, the transcendental produce a mathematics of patterning. He imagination, posits a prior and what J.G. does this by producing a series of pattern Ballard calls archaic state before subject generators, similar to Klir's generative and object arise to give us the duality models that produce data patterns. The and the unsolvable problem. All we have generators exist at the present-at-hand to do is return to the monolith in which level but the patterns themselves exist at subject and object duality does not exist the structural level. In fact we will and our metaphysical problem with differentiate four different kinds of transcendence never becoming patterning called, sign, value, process immanence is solved without sacrificing and structure. It is these kinds of the necessary combination of reason and patterning that produce the experience that is the foundation of transformations in content that underlie understanding. So as we look at the form. These transformations become unfolding of the nomos exposed by deep apparent when we emphasize the mathematics we will track that against dualities between forms and their the models and the logic that stands on opposites and between a pair of the side of logos and compare that to the form/anti-form and everything else. phenomenology and unbracketing or designation as real, identical, true, absent The Greimas Square has the following that takes us out of the form: phenomenological mode. A------non-A : : As we look at the unfolding of the : : nomos we will start with category : : theory. We know that category theory is : : the current foundation of mathematics anti-A------anti-non- A and we know that it allows the commutative property to slip but holds In the Greimas square the contradictory of ‘anti’ is ‘non’ and the contradictory of

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A is anti-non-A which is the Other. A This cubic formation has the form of the and anti-A are contraries as are non-A Octonion. It combines the two Greimas and non-anti-A. We unfold the Greimas book duals into a single self interfering square if we differentiate anti-non-A formation that is chiasmic in all its from non-anti-A. This unfolding is unfolded directions. It is like a glass called chiasmic and represents the cube with warpages in the glass. Each reversibility at the level of Wild Being direction of unfolding holds a chiasmic that Merleau-Ponty calls 'Flesh' in The reversibility and these phase transitions Visible and the Invisible. The unfolded within their respective intervals interfere Merleau-Ponty/Greimas 'book' appears with each other at the center of the cube thus: giving the embodiment of Wild Being as a complex and chaotic phase space. non-A'------A------non-A : : : What we want to do is recognize a : : : : : : similar unfolding of category theory in non-anti-A anti-A anti-non-A which we posit that association no longer holds. This is precisely the There is a dual of this book which has the form: property loss that generates the octonion from the quaternion. We want to apply Anti-A'------A------anti-A : : : this property loss to standard : : : Mathematical Category Theory and see : : : what happens. In other words arrows anti-non-A non-A non-anti-A will no longer be able to be reversed to produce the dual of the category. With These two books may be combined to this loss much of the explicit power of produced the Unfolded Greimas Cube Category Theory is lost so it is no which has the form: wonder this kind of transformation was

not considered before. We must remember that we have both arrows within categories and arrows between categories. We will demand that both kinds of arrows lose their reversibility.

This is similar to the anti-book and the non-book duals of the Greimas square non-A' non2-A unfolding. We suggest the name Proto- / : / : Category Theory for the discipline / : / : which loses association in both arrows / : / : A------:------non-A: and functors. Note that this loss is at the : : : : global level not at the local level. So : : : : there are islands of association similar to : : : : the island where the commutative : : : : property holds despite a global failure. : / non-anti-A : / anti-non2-A : / : / The failure of a property globally causes : / : / a kind of global incoherence which may anti-A anti-non-A be saved locally. The relation between local coherence and global coherence is the dual of the local non-duality verses

8 Deep Mathematics and Meta-systems Theory by Kent D. Palmer the global non-duality of the mobius specific embodiments of order to the strip. Thus the Penrose triangle (Escher seeming disorder of the non-associative waterfall) is the dual of the mobius strip field of proto-categories in order to in some sense. It is the ability to understand the source of the ordering of maintain the property locally in spite of the field that sustains the categories and global failure that produces all the organizes our passage between the wondrous variety of categories and we different categories. Different categories expect it to produce a corresponding are useful for different things. Our variety of proto-categories. thoughts move between the categories looking for templates that will aid our We use the term proto-category as understanding. This movement between opposed to category in order to harken categories encounters the proto-gestalt back to the difference between the which appears in the interstices between gestalt and what I call the proto-gestalt, the categories. There is a netherworld of which is a temporal gestalt. The proto-categories that lack the associative temporal gestalt is what organizes the property that is completely hidden by showing and hiding of gestalts within present-at-hand mathematics. Present-at- manifestation. It has what David Bohm hand mathematics only considers the called an implicate order. In other words islands of the categories not the ocean of there is a hidden order that underlies the ambiguity between the categories that specific gestalts and organizes our we encounter every time we leave the movement from one gestalt to another. safe harbor of a specific category and set In the case of mathematics these gestalts sail for another category in our search are the different categories, like set, for an understanding of all the facets of group, lattice, etc. There is a field of mathematics or in search of prototypes these categories and the categories of order by which to understand some themselves are organized within this phenomena that we are studying. In that field on an unknown basis of ordering. sea there are myriad shadows of the Why do we have these specific categories which haunt the pristine categories with this specific ordering. designated as real categories themselves That is because the categories that exist in which fuzziness and ambiguity reigns are unfoldings of the underlying nomos instead of the clear and distinct forms of in different ways. The fact that there are order that the categories embody. functors between categories shows us that the same underlying patterning First we lose the associative property so mechanism is appearing in multiple that our arrows do not reverse, so that categories in slightly different but the dualities of our categories break recognizably the same incarnations. apart, then the arrows of the different When we move into the field theory morphisms become probabilistic and underlying the categories what we hope then fuzzy and finally chaotic. When we to do is get closer to the source of all lost the commutative property we these similar projections. However that allowed for a much greater range of lies beyond the disintegration of categories to enter mathematics. Now category theory into proto-category when we lose the associative property theory. We must move beyond the and enter the realm of proto-categories gestalts of the categories with their an even greater number of possible

9 Deep Mathematics and Meta-systems Theory by Kent D. Palmer categories become visible. But level of form to the level of pattern in eventually the very morphism lines our consideration of the ordering from themselves begin to move through the the nomos. The emergent ontological process of unraveling as they become template of understanding through forms probabilistic entering into the realm of encompasses forms in space and forms Process Being, then they become fuzzy in time. Forms in time are actions. When entering into the realm of Hyper Being, we loose the commutative property we and finally they become Chaotic allow forms in time to have different entering into the realm of Wild Being. effects when reversed. When we loose Past that chaotic stage of the unraveling the associative property however forms of the morphisms of our proto-category in space also disintegrate because we theory we say that the morphisms cannot cross a boundary and re-cross it themselves become empty because the to get back to the same place. The is the lines of relation that make up our whole concept behind Spencer-Brown's category theory become completely Laws of Form where he delineates that unstable and undependable. At that point crossing and re-crossing a distinction in we reach the limit of our ability to model null. When we lose the associative the nomos. property the context on the outside of the form effects the crossing and re- How are we to understand this limit that crossing. We fall into the dual of the we have reached and the stages that we Laws of Form which I call the laws of have gone through to reach it? First we pattern. In the laws of pattern layering is note that we have step by step unfolded affirmed while multiplicity is denied, the Greimas Square into the Unfolded while in the laws of form multiplicity is Greimas Cube. In other words we know affirmed while layering is denied. The that Category theory is made up of dual two sets of laws are as follows: categories produced by reversing their arrows. These dual categories like set Laws of Form and anti-set are seen in terms of the ()() = () (()) = null Greimas square as A and ~A. The pre- set of the categories are seen as Non-A. But when we lose the associative Laws of Pattern property we find that there is a ()() = null difference between Anti-Non-A and (()) = () Non-Anti-A. In other words when we When we lose the Commutative property look at the gestalt of a single category its in the production of Category Theory we anti-category and the other possible are allowed to create a kind of bridge categories form the background of our between all the different manifestations delimited exploration of a single of nomos. The points of reference vanish category. We know we can get to the in favor of a formalism based on anti-category by reversing the arrows. morphisms. The actions within a We know we can get to the other formalism are allowed to be non- categories by following functors. But commutative. This relaxation of the when the arrows lose their associative basic operation of the formalism admits property we can no longer count on many new formalisms into a designated these morphisms and we fall from the as real status within mathematics and

10 Deep Mathematics and Meta-systems Theory by Kent D. Palmer also allows us to understand the relations allow for other categorical aspects between all the defined formalisms. But within a single category. The duals by relaxing the associative property takes us becoming not immediately reversible out of the emergent level of form all appear as a cloud of different together. It relaxes the associative alternatives that encompass aspects from property. Suddenly our bridges between other categories. So eventually we see categories in the form of functors are no that the non-anti-A is different from the loner reversible. Our dualities between anti-non-A as the variety of the category and anti-categories breaks categories beyond the one in question down as well. Suddenly the context are taken into the relation between the between categories is important and specified category and its dual. This effects a meta-morphism transition opens up a chiasmic field within each between categories. Suddenly the dual category in which the field of categories can break into multiple images of the enters into the category itself producing other as it is infected by different a set of reversibilities with identifiable categories that it intermingles with in the phases in which pure associative background of the gestalt of the within reversibility has been lost so that the which it is a figure. All this brings us phase of association between categorical down to the emergent cognitive level of nodes is differential from the opposite or pattern from the level of form. Since we returning phase of association. The full can not cross out of the form and back logic of this internal separation of the again with impunity and since the Category from its opposite via non- context outside the category suddenly associative arrows and its reflection in matters we are thrown into the opposite the field of all the categories to which it of the laws of form where the nesting of is intrinsically related via non- categories suddenly matters as in the associative functors is embodied in the laws of pattern. In this state our Unfolded Greimas Cube. That cube has categories gain depth as does the field multifarious reversibilities which all between them. Different routes though interfere with each other to produce the the landscape of categories yield image of Wild Being. Pure Presence different results and paths across the Being is the level at which there is a landscape cannot be reversed easily. We clear delineation between a category and begin to see categories within categories its opposite and with the field of all the within categories as the nested other categories on which it is defined. categories reflect the whole field of Process Being appears in the movement categories. Suddenly the field transforms of the morphisms itself either within a into an interpenetrating jeweled network category or between categories. Hyper in which every embodiment of order Being appears in the non-decidability reflects the rest of the embodiments of between a category and its anti-category, order. All the functors become or between two categories within the internalized as reflections of other field of possible categories. Wild Being categories within the field. Since we appears suddenly when we drop the cannot traverse the field with impunity associative law that links a category to any longer we switch to seeing the its opposite or between categories within mirrorings of the field within the the seascape of island categories. Wild individual category which is deepened to Being appears in the chiasmic

11 Deep Mathematics and Meta-systems Theory by Kent D. Palmer reversibility that is not associative until we get to the point where all the between a category and its dual and the difference is lost between the categories nesting of other categories within that and we enter a pure fusion of all the dual OR between a category and another categories with all the other categories category in which all the other categories and our ability to distinguish between interfere. This move toward Wild Being them and divide them from one another in mathematics is recapitulated in the is completely lost. You would think that indeterminacy of the arrows themselves. below the level of pure presence which In fact what is occurring is that not only normal category theory enjoys there do the categories undergo the opening up would be nothing but vague chaos and of chiasmic reversibility but the arrows indeterminacy. But in fact there is undergo a similar process except in their gradual shading off of order and that case it is expressed in the arrows losing fading of order is part of the nomos itself their purely present determinism. The as well. In fact we find an important arrows enter into Process Being when aspect of the nomos here which is its they become probabilistic. They enter non-duality. So that is why it is into Hyper Being when they become important to us to explore the field fuzzy. They enter into Wild Being when within which the categories arise and the they become Chaotic that is when they transitions between categories as well as exemplify a mixture of order and just the present-at-hand categories disorder. themselves. It gives us some understanding of the deeper levels of the Now we might ask why is it that the nomos beyond the category theory and it opening up of proto-category theory has gives us some comprehension of the this specific kind of structure rather than underlying structure of our world. some other. Why must we lose the commutative property in order to Now in order to understand why the establish the categorical imperative deep structure underlying mathematics which links all the possible categories has the form that it does we must delve and gives us a universal mathematical into the distinction already breached language for comprehending them. But between a System and a Meta-system. then what happens when we go on to We have breached it when we talked lose the associative property in our about a category as being a gestalt and transition to the underlying field that we contrasted that to the proto-gestalt supports the categories. Finally the which we enter when we lose the arrows lose their definiteness and associative property. When we do that disintegrate though a series of similar we enter the field between the categories steps taking us though probabilities, themselves and that field itself becomes fuzzification and finally into chaos. The mirrored and reflected within the reason behind this has to do with the individual categories. In other words, the way we construct our world. Our Indo- functors become internalized within the European world is constructed in layers category and the dual category becomes and when we explore the basis of differentiated by those reflections so that category theory we are opening up the like a hologram the field of categories is deeper layers of Being within our world. introjected into each category within the We peal back these layers one at a time field. Proto-category theory becomes a

12 Deep Mathematics and Meta-systems Theory by Kent D. Palmer proto-gestalt in the sense of a temporal different paths and situate their gestalt. In other words our path of specialties with respect to different thought between the categories becomes prioritizations of the categories important suddenly and by taking themselves. The mathematical journey different paths between the categories directly confronts the proto-gestalt. we end up in different places. Going out What we see when we come to set to another category and coming back theory from lattice theory is completely places us in a different context rather different from what we see when we than exactly where we started. Looking come at it from group theory. All the at the reflections of other categories in a various categories can be seen though specific category in different orders the mediation of the others and that changes our comprehension of that mediation changes our comprehension of category. At the associative level the the categories which is our specialty. dynamics of our exploring the landscape of the nomos as it is represented in the The purely present mathematical results field of categories becomes extremely that appear in articles and books that significant. We know that different delineate pristine and pure formalisms mathematicians have different view hides the actual process of manifestation points. What we have not considered is by which mathematicians undertake their the fact that this is because they have studies and approach their results. In that explored the landscape of the nomos in process of thought they wander though different orders and the viewpoint they the proto-categorical landscape and have constructed is an artifact of this attempt to keep in mind the purified and journey that they have undertaken in pristine forms of the categories and their studies. Viewpoints are produced as distinguish them from that ocean of articulations of the field of the proto- confused resemblances. When we find gestalt. The social cohort of living and discontinuities within that landscape interacting mathematicians exemplifies where resemblances fail and new the proto-gestalt in the concretization of categories have to be created then we their different mathematical viewpoints. have encountered Hyper Being. When They know in practice that there are we realize that the discontinuities and different ways of relating categories and the continuities are fundamentally mixed relating to categories and they carefully up in our representations and that the attempt to hide that in their publications underlying nomos that produces the which focus on present at hand results, differences and similarities is non-dual stripped of the process of discovery and then we enter the realm of Wild Being. the meanings that attend to their When we loose our ability to distinguish comprehension of the nomos. Each even on the level of pattern after we mathematician has to travel through the have lost our hold on form completely landscape of the categories and those then we enter the realm of utter non- journeys are all different depending on duality and emptiness where the nomos the paths taken in each specific case. It is becomes unthinkable completely. not just because mathematicians are different that there is such variety of All this exploration of the underlying mathematical viewpoints created. It is basis of mathematics that seeks to look because they traverse the landscape in into its ungrounded foundations is

13 Deep Mathematics and Meta-systems Theory by Kent D. Palmer dependent on a simple model that break down and become both underlies the articulation of our Indo- indeterminate and become non- European worldview. It is expressed in associative then we enter fully into the the following figure: meta-systemic proto-gestalt that the proto-category theory manifests. This is SYSTEM the field out of which mathematicians Pure Presence Being DISSIPATIVE SPECIAL SYSTEM complexnion attempt to wrest the discrete and clear Process Being categories that they hold in pure AUTOPOIETIC SPECIAL SYSTEM quaternion presence and display in their papers and Hyper Being REFLEXIVE SPECIAL SYSTEM octonion books. But each paper and book is a Wild Being narrative that winds its way though the META-SYSTEM field of mathematical forms in a certain order. The ordering of the journey When we speak of the gestalt and proto- effects deeply the results that appear gestalt as a way of understanding the along the way. Ultimately we can see the difference between associative category categorical landscape to be very much theory and non-associative proto- like a jungle of possible orderings and category theory, then we are making in a each mathematical paper or book cuts a different way the distinction between a specific path through that jungle seeing 'system' and a 'meta-system', or what intimations of underlying order Bataille calls a 'restricted economy' and everywhere but differently than would a 'general' or 'global economy'. The have been seen on some other path. meta-system is the environment, context, or situation that surrounds and What is important to understand is that encompasses a system. I define a system between the system and meta-system, or as a social gestalt from the point of view between the gestalt and proto-gestalt of social phenomenology. A meta- there are several specific steps that are system is the social situation or social defined as the special systems and called context that surrounds the seeing of a dissipative, reflexive and autopoietic. socially designated as real, true, identical Where a system is a gestalt that is a or present gestalt from a particular whole greater than the sum of its parts, socially constructed viewpoint. The i.e. having emergent properties, a meta- meta-system can be seen either as the system is a proto-gestalt with implicate side-effects or shadows of the system as order that is a whole less that the sum of it haunts other systems within the its parts , i.e. having de-emergent lacks. categorical field, or as the fragmentation The special systems are defined in terms of viewpoints that cast those shadows. of algebras so that the dissipative special Each category is a restricted economy system is ordered by the algebra of conforming to certain mathematical complex numbers, the autopoietic definitions. But each category special system is ordered by the algebra participates in an underlying categorical of quaternions, and the reflexive special landscape that forms a general economy. system is ordered by the algebra of The functors are a way of talking about octonions. These algebras are dependent this underlying similarity that comes on properties and lose properties as we from the non-dual nomos expressing go down the cascade from the system to itself in various categorical the meta-system. The dissipative special embodiments. But when the morphisms

14 Deep Mathematics and Meta-systems Theory by Kent D. Palmer system with its complexnion algebra has When we look at category theory we see all the properties of the real numbers that that it allows the commutative property define the system gestalt. The to relax so that it takes us to the level of autopoietic special system with its quaternions in order to build a bridge of quaternion algebra loses the functors between categories and develop commutative property. The reflexive a universal language to describe all the special system loses the associative mathematical categories. It defines the property. The meta-system is associated morphism as something determinate and with all the other non-division algebras purely present. The elements vanish in that lose the property of division such as category theory the morphisms become the sedenion which is generated by the arrows that represent transformational Cayley Dickson process as the next processes in order to build a bridge possible on beyond the octonion. But the between the forms of order represented infinity of non-associative algebras by the categories. But the arrows that associated with the meta-system are of represent morphisms are reified and little interest to us because they do not remain present-at-hand. If we seek to allow the intertranformability of deepen category theory that establishes numbers by algebraic manipulation that itself on the structural process level we we find so useful and that attract us to need to release the associative property these three unique division algebras. The to become relaxed as well and then point is that in the transition between the release the division property. At the system gestalt and the meta-systemic same time we need to allow the arrows proto-gestalt we encounter these specific to become indeterminate moving though algebras as we lose algebraic properties a progression from probabilistic, to one by one. The difference between fuzzy and finally to chaotic before losing these steps define the kinds of Being. So our ability to differentiate them. This as we are traversing the series of steps progression down though the layers of we are also moving from one kind of the kinds of Being of the world and their Being to another until we get to the associated special systems merely meta-system that represents pure completes the journey that category existence outside of any hint of Being. theory itself begun. Category theory is a That existence is unthinkable and can be structural theory which attempts to interpreted as the Buddhists do as explain the differences between sunyata, i.e. emptiness that is itself categories and bridge those differences empty. The kinds of Being define the by constructing a language of differences between the special systems morphisms that can be universally as the special systems define the applied across all categories. At the differences between the kinds of Being same time it relaxes the commutative in a very deep reciprocity. It is this property in order to allow in all the non- reciprocity that defines the underlying commutative categories. Now we insist structure below category theory which is that it is necessary to relax the poised at the level of pure presence in associative property to allow in all the some ways an process being in other other non-associative proto-categories. ways. When we do that we take ourselves down into the level of Being called Hyper Being. At this level of being there

15 Deep Mathematics and Meta-systems Theory by Kent D. Palmer is indeterminacy between various category theory not in order to destroy positions. This indeterminacy opens out mathematics but in order to open up a its indecisiveness into an articulated deeper mathematics that is concerned field that appears only when we cannot with the non-dual nomos directly reverse arrows at will. Duals are no embodied rather than merely represented longer interchangeable. The dual is by the reflections of the categories in invaded by the background of the other each other. We will use a series of categories. Functors become paths of mathematical analogies in order to thought between categories that cannot explore this non-dual dimension more be easily reversed and allow the deeply. interembedding of categories. After releasing the associative property we Between the logos and the physus find ourselves at a point where we must appears the non-dual nomos. Logos is release the division property as well so articulated as truth and identity that it becomes impossible to distinguish expresses logic as the move from truth to categories any longer as their mutual identity. Physus is articulated as reality mirroring overwhelms our capacity to and presence and expresses the testing of differentiate. This occurs at the same scientific experiment as it moves point that the arrows after becoming between the two. Reality is the filtering probabilistic, fuzzy and chaotic and testing of the system which is themselves become indistinguishable. At presented to the meta-system. Truth is that point we reach the limit of our the relation of the system to the meta- capacity to differentiate the nomos. On system which attempts to remain true to the way to that liminal mathematical itself in order to avoid meta-systemic state we unearth the non-dual nature of filtering. So when we distinguish the nomos underneath our projection of between the system and the meta-system the categories as a structural way of we are articulating the spit between comprehending forms of order. It is that logos and physus and the split between non-dual form of ordering that underlies the aspects of reality they encompass. all the dualistic projections that we want All this exemplifies the limited (peiron) to explore more deeply. Realizing that as distinct from the metaphysical the Unfolded Greimas Cube of principle of the transcendent and reversible chiasmic relations exist within unlimited which is Being itself as a the categorical landscape is a key point whole (Apeiron). Between the limited in the evolution of our mathematical and the unlimited is another non-dual intuition. Where category theory is moment called the RTA (ASA) which poised at the quaternion level we know today as the RIGHT but which algebraically the Unfolded Greimas used to have the meaning of ‘cosmic Cube of non-dualities is located at the harmony’. There is a difference between octonionic level which is reflexive and knowing what should be done, i.e. the inherently social. This is why we nomos, and what is right in any develop a social phenomenology to situation. This difference appears outside attack that level of the articulation of the of mathematics. Plato talks about it in proto-gestalt. We define the limits of terms of the divided line as the mathematical thought in the difference between mathematically disintegration of the purely present distinct intelligibilities and

16 Deep Mathematics and Meta-systems Theory by Kent D. Palmer intelligibilities that cannot be described differentiated by the different kinds of mathematically. When we deepen our Being. Beyond the cave is existence mathematics we move beyond the which is empty and thus interpenetrates. nomos to the level of the source of the The Good is the source of variety in Beautiful and on to the source of the existence and thus of the variety of the Good. The source of the Beautiful is the categories. The Good is a cornucopia of same as the RTA or right. In the variety production which we see at work Republic Plato deals with rights in terms in the industry of mathematical of justice. Justice is seen as the definition of ever new kinds of distribution of rights among individuals categories and elaborations of existing within the city. The best distribution of categories. When we move deeper into rights is the most beautiful. In terms of the Nomos we encounter its limit when the Parable of the cave there are the it reaches the Beautiful (outwardly) or objects held by the Sophists which are Right (inwardly). Beyond that is yet seen by the bound spectators as another deeper non-duality which is appearances. The objects and the called the Good which is the source from appearances are the lower echelons of which all the variety comes whether the divided line. But the upper echelons beautiful or ugly or as Socrates says can be seen in terms of the Fire that about love in the Symposium “neither”. allows the objects to be seen and the We see RTA in our will to power Sunlight beyond the Cave. We already seeking elegance, simplicity, beauty and know from the analogy of the Sun that aesthetic appeal in our mathematical and the Good is thought of as the Sun. So we physical theories or formalisms. The can think of the source of the Beautiful non-duality of order looks across to a that we are told to contemplate in the deeper non-duality of Beauty and Symposium as the Fire within the Cave. aesthetic elegance which we find fitting It is the non-dual source of Right. What and which provides our appreciation of is right is in harmony with the cosmos. mathematical order with some degree of What exemplifies harmony is the most ecstasy. Beyond that non-duality we beautiful. Thus rightness has an inner wonder at the variety of mathematical relation to beauty. The source of forms. If God did create with rightness and beauty are the same. It is mathematics as his foundation then we symbolized by the fire within the cave find ourselves in awe not just of the that allows us to see the appearances of variety of created forms but at the the objects held by the Sophists. This is variety of the sorts of nomi that exist. why in mathematics we aim for We can only discover these deeper non- simplicity, beauty and elegance because dualities if we go down into the that is what feels right to our uncertain realm of the proto-categories. mathematical intuitions. It is in that We must enter the self-interfering non- intuition that the Arte (related to RTA) dual chiasms of the Unfolded Greimas or excellence of Mathematics inheres. Cube in order to find the traces of these But beyond the Arte by which we intuit deeper non-dual sources that give our the beauty and rightness of mathematics mathematical skills and intuitions their there is the source of the Good. That reason for the continual exploration of source is beyond the Cave of Being and the landscape of the categories. It is the its projection machinery that is reflections of a deeper order that we see

17 Deep Mathematics and Meta-systems Theory by Kent D. Palmer expressed in the categories that keeps us dual surface with the added feature of searching the landscape of mathematics enclosing space and that the same side and sends us out on the oceans of that encloses space is that which forms thought that separate the categories form the outside of the bottle. We normally each other. What we need is a way to hear more about the Mobius Strip than approach that deeper mathematics that the Kleinan bottle. But Steve Rosen has unearths the non-dualities more directly. done a philosophical study of the both the Mobius Strip and the Bottle which shows that it has deep implications for Non-dual Mathematical Anomalies out thinking and especially that it is a good analogy for non-dual thinking. From the great field of category theory Here we will elaborate on his results. and its opening up into Proto-category One of the most interesting things he theory as the environment of says is that the Bottle is somehow mathematical thinking we are going to between the third and fourth dimensions. focus now on a few mathematical It is this concept that we will explore anomalies that give us intimations of the more deeply in our exegesis. underlying depths that we have described in general terms. These anomalies give us a view of the non-dual Dissipative Special Systems underpinnings of the nomos within We will start by noting that the dual of specific mathematical contexts which in the Mobius Strip is the Penrose Triangle. themselves are dualistic and present-at- The Penrose triangle is that impossible hand. We will be focusing on the three dimensional looking drawing of a anomalous construction of the mobius triangular configuration of slats that strip within geometry and extending that appears to turn inside out as we follow it to higher dimensional geometrical around becoming an impossible figure. constructions. In general the mobius Escher used this triangular configuration strip is a precise icon of non-duality due to Penrose effectively in his because it is locally dual and globally Waterfall Etching. Generally a Penrose non-dual due to the fact it as only one triangle is an impossible paradoxical side and one edge despite its appearance looking figure which makes sense when of being two sided and two edged when you look at each vertex but does not looked at narrowly. In the mobius strip make sense when it is looked at globally. the one edge wraps around itself to Such a figure cannot really exist in three become both edges of the strip and dimensions. The difference between the similarly the single side wraps around it Mobius strip and the Penrose triangle is to become both sides simultaneously. that the Mobius strip achieves non- This occurs by the strips twisting in duality in reality while the Penrose space and is in fact composed of a single triangle is only an illusion of a figure strip of paper twisted once with its ends that wraps around itself but which joined. Mobius strips can be twisted cannot exist in three dimensional space. either left or right and when two The Penrose triangle has the property of oppositely twisted strips are joined global incoherence with local coherence. together then form what is called a Each vertex appears normal but the Kleinian bottle. The bottle is also a non- global configuration is skewed

18 Deep Mathematics and Meta-systems Theory by Kent D. Palmer impossibly. The difference between means that in anomalous far from local coherence and global incoherence equilibrium chaotic systems it is possible is the opposite of local duality and for order to be produced. That order global non-duality. The non-dual is dissipates outward from the center coherent but in such a way that duality is toward the periphery. Such systems are embedded within it. Local coherence has neg-entropic in a local area. Globally to do with the configuration of the entropy is maintained but in rare aspects of the pictured object. When the conditions local order generation is object is looked at as a whole it lacks possible. The order appears from a coherence even though the individual singularity and move out toward the parts seem to fit together correctly. What boundary of the dissipative special the Mobius Strip tells us is that we can system. At the boundary the dissipative have non-duality without losing special system disorders the environment coherence globally. Also what the and reorders it according to the new Penrose triangle tells us is that losing ordering regime flowing from the global coherence comes from the negentropic center. Because the inability of the parts to fit into the whole boundary is always bigger than the properly. The Penrose triangle is a figure singularity that means the total entropy that does not have holonic properties, in is always greater than the order created other words the local configuration does so the general law of entropy is not fit into the global configuration. On maintained. A good model of the the other hand the Mobius strip has a Dissipative Special System in nature is peculiar integrity that allows for the two the Soliton wave. Such a wave maintains sides to fit together so well as to be one its energy much longer than one would side and the two edges fit together so normally expect. Because of this we well as to be one single edge. Integrity consider the soliton super efficient. Such and Holonic characteristics are very high a wave is like a dissipative structure. The on the scale of harmony that Cheng boundary is the trough within which it produced where mutual support is the travels. The wave is reflecting off the highest level prior to interpenetration. walls of its trough and continually Cheng's levels are strife, logical relation, reconstituting itself from those mutual interaction, mutual support and reflections. The anomaly of the super- interpenetration as we move up toward efficiency of the soliton wave is off set higher and higher levels of harmony. by the energy necessary to create the Mobius Strips exemplifies mutual tough that it must live within. But in the supporting integrity of a peculiar non- soliton we see the continual circulation dual sort. The Penrose triangle lacks of reflected energy from the soliton to holonic mutual support of parts fitting the boundary tough and back to the into the whole. center as it moves down the trough. As it moves it will go right though another The Mobius Strip and the Penrose soliton without losing energy, it will turn triangle relate to the dissipative special corners with the channel that it is system. That special system has been following and bounce off walls also described by Prigogine in Order Out of without losing energy. Soliton solutions Chaos. He calls dissipative special exist for many physical equations so systems 'dissipative structures'. This their possibility is rife in nature.

19 Deep Mathematics and Meta-systems Theory by Kent D. Palmer

Similarly we can say that the Mobius it does not mean that it is impossible strip is an analogy for the dissipative with information. Information flow special system as well which can be based on consumption of energy may modeled mathematically by complex cause negentropy or some other super- numbers. Complex numbers have an odd efficiency when it takes the form of the twist out of the plane of the real numbers Mobius strip becoming like a soliton when we take the square root of negative with the underlying mathematical one. The Mobius strip models the structure of the complexnions. twisting around at the boundary and the Dissipative structures are perpetual bouncing back of energy to the information flow machines. They take singularity where the order is composed advantage of the fact that chaotic and seemingly flows in from nowhere. systems can have infinite information as But another interpretation is that there is in strange attractors and use energy to a potential well between the boundary promulgate the information flow in ways and the singularity where the disorder of that appear and actually are super- the boundary twists back into the efficient. Dissipative Special Systems singularity to become the order that are the systems theoretic prototypes for emanates from the singularity. The all such super-efficient perpetual Mobius strip shows us how the information machines. Although negentropic ordering can be the flip side perpetual motion machines are of disordering so that the same process impossible perpetual information that disorders the environment can be conversion machines are not. This reflected in the ordering within the difference between possibility and dissipative special system. impossibility is reflected in the difference between the Penrose triangle Although dissipative special systems are and the Mobius strip. If the waterfall of negentropic they are not perpetual Escher contained information instead of motion machines. They do not create water or energy then it would be energy nor do they balance energy. What accurate. The point is that even though they do is consume energy but in such a each individual conversion of the way that spontaneously produces order perpetual motion machine might make in a determined small region of the sense the global incoherence of the world which feeds on the disordering of machine makes no sense because it larger regions of the world. The violates entropy laws. However, perpetual motion machine almost always information is disembodied in a way has a form similar to the illusory form of such that for information these violations the Penrose triangle. In the perpetual do not occur. Information transfer uses motion machine an impossible flow is energy and expends it but the set up to try to recirculate energy information itself can be conserved or impossibly back on itself somehow. We produced by algorithms super know that there is no perpetual motion abundantly so that information seems to machines just as there are no three come from nowhere. The difference dimensional Penrose triangles. Both are between the illusion of the Penrose a trick that is really impossible to pull triangle and the Mobius strip is off. But just because it is impossible to instructive because it is through non- pull off that trick with energy and matter duality that it is possible to produce this

20 Deep Mathematics and Meta-systems Theory by Kent D. Palmer effect of super abundant ordering within fourth dimension. However, he says that a small portion of space based on energy this makes the actual neck of the consumption. The perpetual motion Kleinian Bottle disappear and turns it machine fails to achieve that nonduality into something else. That something else and thus loses global coherence which turns out to be the pentahedron in four the Mobius strip achieves. One is a dimensional space, i.e. the analog of the carefully constructed illusion and the tetrahedron which is the simplest other is a realized reality. polytope in that space being composed of five orthogonal points, ten lines, ten triangular sides and five tetrahedrons Autopoietic Special Systems combining into one four dimensional figure. The sides of the pentahedron Once we understand the relation have two sets that can be reduced to two between the Mobius strip and the intertwined Mobius strips. The Penrose triangle and their relation to the pentahedron has no anomaly of self- dissipative system that can be said to be intersection. In four dimensional space analogous to the soliton and the pentahedron is a balanced figure with complexnions we are ready for the next no asymmetries unlike its three step. Each of these figures may be dimensional analog the Kleinian bottle. I related to a further development. Two believe that Steve Rosen has had a deep Mobius strips produce a Kleinian bottle. insight when he says that the Kleinian Similarly there is another paradoxical bottle is non-dually between dimensions. figure called the Necker Cube which has It is both the Kleinian bottle and the similar but slightly different properties pentahedron and its self intersection than those of the Penrose triangle. What point is key to understanding non-duality is interesting is that when we go up to as it manifests in our world. The point of this new level the properties of the two self-intersection is like the concept of a types of figures are swapped. The set being a member of itself. Onar Aam Necker Cube is globally dual and locally pointed out that the set and the Kleinian non-dual while the Kleinian Bottle is bottle have similar characteristics and I globally coherent and locally incoherent. am extending that analysis with respect Notice this switching back and forth to the Hyper-set or the Non-Well- between the properties of the two dual founded Sets of Aczel. Such anomalous figures. The Kleinian bottle is globally a sets can be members of themselves or coherent figure but locally it has its self- self intersect. When we look at the intersection where it is incoherent. With global duality of the paradoxical figure respect to the Necker cube we find that of the Necker cube and compare it with globally there are two ways of looking at the Kleinian bottle and its embodiment the cube that oscillate in our perception. of local incoherence in space, we are But locally this dual oscillation vanishes lead to consider Riemann space that is when we look at a single vertex. Steve globally incoherent but locally coherent Rosen says that the Kleinian bottle is which is more like the Penrose Triangle. non-dually between the third and fourth Riemann space is the opposite of the dimension because we could make the Kleinian bottle. When we look at the self-interference disappear if we moved bottle we see that it is a single surface the neck out of the third dimension embedded in a cast of space. Since the either toward 'anna' or 'kappa' into the

21 Deep Mathematics and Meta-systems Theory by Kent D. Palmer bottle is asymmetrical and has two Mobius strip and the Kleinian bottle are different ends there are four possible examples of super-rational non-duality. bottles and thus four possible casts of Notice that the former is always an space around such bottles. We can illusion while the latter is always a consider the space within which the realizable anomalous reality. One of the Kleinian bottle appears to have global key distinctions on the road to wisdom is curvature like the Riemann space and the difference between the super-rational thus have a fourth invisible dimension. and the paradoxical. The latter represents The global curvature is invisible to us intellectual sickness while the former but on the other hand the ambiguous represents embodied health at a deep space of the self-intersection is very level. Our figures tell us how closely visible. The Kleinian bottle is a single intertwined the two enfoldings of reason surface but it articulates a three space. can be. One takes us into illusion while Thus it is two dimensional and three the other takes us into a reality that is dimensional at the same time in different highly integral. The super-rational aspects. Riemann space is four combines the holonic and the integral dimensional and three dimensional at the into the interpenetrating. The same time in its coherent and incoherent paradoxical takes us into the strife of aspects. The Necker cube is a three opposites and mysterious conjunction dimensional illusion but actually must that has no reality. inhabit only two dimensional space of representation. The Necker cube is an The Kleinian Bottle and the Necker cube illusion while the Kleinian bottle is a also relate to the next higher special realizable reality. system called the autopoietic. The autopoietic is a conjunction of two This brings us to the consideration of the dissipative systems. It abides by the difference between paradox and super- mathematics of the quaternion and is rationality. Paradox is a sickness of the exemplified physically by intellect which creates an illusion of superconductivity. It is more than super- sophistication. Super-Rationality is non- efficient it is in fact ultra-efficient and dual and produces a reality that goes violates local entropy completely. beyond dualistic constructs. The Autopoietic systems were defined by Western tradition is fascinated with Maturana and Varela. They are systems paradox, as we see in Hofsteader's Godel that produce themselves and hold their Escher Bach: Eternal Golden Braid for organization as a homeostatic variable in instance, but has little experience or spite of structural changes below the understanding of super-rationality. threshold of their organization. Maturana Super-rationality was developed to a and Varela's theory is really about refined level sophistication in Buddhism. autopoietic forms and that is defined by We see the super-rational at work in Zen an appeal to the structural level of Koans. In the super-rational we hold two patterning below form and the opposite positions at the same time organizational or systemic level of without conflict. In paradoxicality there holistic ordering above the level of form. is conflict between the opposites. The Spencer Brown describes these Penrose triangle and the Necker cube are autopoieitc forms very well in Laws of examples of paradoxicality while the Form. They are forms in which the

22 Deep Mathematics and Meta-systems Theory by Kent D. Palmer operator and the operand are the same. An autopoietic system has a set of nodes Such a form Spencer-Brown calls a that produce itself. These nodes are “mark” in his philosophical boundary controlled by a hyper cycle set in four algebra. Maturana and Varela's theory dimensional space away from structural magically introduces observers at the network of nodes. We can define an system level and so violates coherence autopoietic system by noting that there defining life as a paradox. We instead are four kinds of patterning: value, sign, like to think of life as non-dually super- process and structure. The former are rational and prefer to define it by first defined by Baudrillard in Critique of the distinguishing systems from meta- Economy of the Sign and the latter are systems and then distinguishing the three defined by George Klir in his chiasmic special systems as a means of defining epistemological hierarchy (Architecture the autopoietic system as the balance for Systems Problem Solving). When we point between the dissipative and move up from the level of pattern where reflexive. Maturana and Varela use structure as the changing substrate of the autopoietic We see in the Kleinian bottle the form we find that we can delineate the definition of inside and outside by the boundary of the autopoietic form using balance of two opposite Mobius strips. boundary algebra of the Laws of Form. This is precisely the same as the This algebra allows us to delineate the balancing of complexnion vectors in the form of the autopoietic system. However quaternion. However, when a symmetry it ignores the “Laws of Pattern” which is breaking occurs the two complex vectors the dual of the Laws of Form. Both become a single quaternionic object that Form and Pattern are needed to define embodies the twisting of four the autopoietic system. The Laws of dimensional space. Such twisting is Form is a model of transcendence which hyper-efficient in that it prevents all ignores immanence. Both transcendence knotting or blocking interference. and immanence need to intersect in the Mobius strips conjunct to form a Autopoietic form. What Stafford Beer Kleinian bottle and they have the calls the “muddy box” needs to be taken property that the inside and the outside into account as well as the control surfaces are the same as well as being structures that attempt to control it but non-dual. There is no edge in a Kleinian ultimately cannot. We assert that it is bottle yet it defines a boundary with necessary to use the Surreal Numbers as what lies beyond it. It also bridges via a model of the bifurcating structure that the circle of self-intersecting ambiguity is projected down on the patterning and into the fourth dimension. At the circle to serve as content at the level of of ambiguous space the figure is at once abstraction where the Laws of Form are open and closed to the outside world. It poised. We wish to include the “Laws of is the very definition of an openly closed Pattern” as the dual of the Laws of Form system. Such a system has access to the which emphasizes the importance of outside world without breaking its nesting over multiplicity. These two boundaries. In the Kleinian bottle this exclusive viewpoints are like the duality ambiguity is embodied. of the Necker cube, their separation is illusory. We really need an algebra that includes both Form and Pattern. This can

23 Deep Mathematics and Meta-systems Theory by Kent D. Palmer be achieved by combining the Surreal operations occur in the rings that August numbers of Conway as the abstract Stern describes they are autopoietic. The content of the Laws of Form and also by Matrix Logic describes the designation realizing that there are also imaginary as Real, as True, as Present, as Identical forms as well as real forms to contend of the Autopoietic System with respect with. We signify imaginary forms on the to the meta-systemic environment. pattern of the quaternion by rotated marks. By that we situate the The autopoietic system appears like a Autopoietic form in a four dimensional paradox when it is defined as Varela and milieu. That four dimensional milieu is Maturana do as the intersection between differentiated into the spacetime of systemic organization and patterning process and structure and also includes a structure. But if we instead use the hyper-space of sign/value in which the distinction between the meta-system and hyper-cycle occurs which controls the the system and go on to define the self-production. The best model of the special systems then we realize that the hypercycle is the five Hsing autopoietic system is better represented (transformations) of the Chinese. We as something non-dual like the Kleinian note also that the model of the human bottle only dynamic. The definition of body used in Acupuncture is basically the autopoietic system as Maturana and autopoietic. They explicitly model the Varela do is like the Necker Cube. It body as a series of channels in which relies on nihilistic opposites for the solitonic Jing Chi flows which is self- definition which oscillate undecidably maintaining and self producing. Within and then it jumps from the level of the four dimensional realm of the Matrix autopoietic form to autopoietic system of spacetime and timespace we see the magically by the introduction of interaction between the controlling observers from nowhere. In reality the hypercycle and the nodes of self- autopoeitic system is itself reflexive and production which in the case of it mirrors the social reflexive realm Acupuncture are the points. within itself. As Onar Aam acutely pointed out on the Autopoietic Email Finally we realize that each of the List the very concept of 'organization' is imaginary distinctions can be made with social. We cannot really separate the respect to the four aspects of Being: dissipative and reflexive moments of the reality, identity, truth, presence. We autopoietic system. It is a conjunction would use August Stern's Matrix Logic between the dissipative and reflexive. as the means of representing the organizational level of the autopoietic system. Each of the fundamental aspects Reflexive Special System of Being may be used as a designation for the ordering of the organizational The next step up in our progression is level. Thus we would have binary truth the Hyper-Kleinian Bottle and the vectors in Matrix Logic for each of the Tesseract. The Hyper-Kleinian bottle is aspects of Being. Logical operations the conjunction of two Kleinian Bottles. would operate upon these truth vectors, The Tesseract is the four dimensional reality vectors, identity vectors, and analog of the Penrose Triangle and the presence vectors and when these Necker Cube. The tesseract is an actual

24 Deep Mathematics and Meta-systems Theory by Kent D. Palmer four dimensional polytope. It has eight four dimensional space. To simplify the three dimensional cubes all linked exposition we will create a construction together though the fourth dimension. in three dimensional space that is This figure is a reality in four equivalent to the Hyper-Kleinian bottle. dimensional space and thus breaks out of We do that by first noting that there are the illusion which the Penrose Triangle two basic representations of the self and Necker Cube incline toward. intersection of the three dimensional However, we run into a different Kleinian bottle. One is the normal problem at that level, which is that it is picture where the neck tube comes impossible to visualize directly the through the side and joins up with the rotations of the Tesseract. Four hole in the bottom. But another dimensional space causes geometrical completely different formation makes intuitions to fragment though lack of the self intersection a circle that goes experience of that dimension. We can round the whole bottle and makes the describe these figures algebraically but it bottle look like a strange kind of torus. is very difficult to reason about them In this formulation we take a figure eight geometrically. Thus, we can see the tube and rotate it 180 degrees and join shadows of the Tesseract as it moves the ends to form a Kleinian bottle. If we though our space, we can see the three rotate the figure eight tube 360 degrees dimensional impression it makes within instead we get a Mobius strip. This our space, but we cannot see the actual formation produces a simpler and more relations between the cube that are elegant form for the bottle and Mobius connected in intersecting three-spaces strip. Now in order to get a Hyper- that make up the tesseract. In this way Kleinian bottle instead take a four leaf the tesseract continues the progression of clover shaped tube where the four the Penrose Triangle and the Necker hollow leaves all intersect in a single Cube. The tesseract as the four line, then twist it 180 degrees to get a dimensional analog of the cube, casts its pair of Kleinian bottles wrapped around shadow on our three dimensional space each other with the same circle of self but the figure itself is out of reach of our intersection. If we rotate the same four experience. Four dimensional space is leaf clover tube and join up the ends incoherent when looked at from the third with a 360 degree twist we get dimension and geometrically but from intertwined Mobius strips instead. When the viewpoint of its own dimensionality we move to identify the four it is algebraically coherent. Incoherence dimensional analog of the intertwined becomes a matter of viewpoint and the Mobius strips our attention is draw to the underlying formalism one chooses as a pentahedron in four dimensional space. basis for reasoning about the fourth The pentahedron is the analog of the dimension. Here we see that incoherence three dimensional tetrahedron in four again returns at this level to be the space. This geometrical figure is two fundamental property that is exemplified Mobius strips intertwined without self by the tesseract as it was by the Penrose intersection. In four dimensional space Triangle. there is no need for the there to be any self-interference between the Mobius The Hyper-Kleinian bottle is a strips in order for them to form a conjunction of two Kleinian bottles in complete figure. The pentahedron may

25 Deep Mathematics and Meta-systems Theory by Kent D. Palmer be seen as inscribed in a Hypersphere inscribed within a sphere. Two such and in that Hypersphere there may be tetrahedrons one from each pentahedron two dual pentahedrons inscribed in may be so oriented so that they are which the center of the faces of each are inscribed in the same sphere. If we look the points of the other. When we at the tetrahedrons we see that each is imagine this composite figure we are composed of six lines. Three of these looking at the analog of the Hyper- lines in a kind of three dimensional Z Kleinian bottle. That bottle is produced pattern belongs to each Mobius strip. from four Mobius strips that are When we look at the two Mobius strip conjuncted along their lines to form two lines we see that what ever tetrahedron Kleinian Bottles and these two Kleinian we choose within the pentahedron will Bottles are conjuncted along their circles become the sphere of ambiguity. When of self-intersection to form the Hyper- we align the two tetrahedrons so that the Kleinian bottle. That bottle may be seen spheres they form line up we have as two pentahedrons arranged in defined a three dimensional sphere complementary positions within a hyper- within the hypersphere that contains the sphere. Or it can be seen as we have two pentahedrons. That sphere of imagined it as two Kleinian Bottles ambiguity can be seen as composed of intertwined except that in Four the two independent circles that form the dimensional space each occupies a four- hypersphere. One circle is made up of dimensional half-space (anna and the x-y plane and the other of the z-w kappa) on either side of a three plane. We can see ourselves as dimensional space that slices the fourth dynamizing these two unit circles and dimension in half. In that three tracing out the two Kleinian bottles dimensional space is a sphere where the independently and simultaneously within two Kleinian bottles self-intersect. The the hypersphere. Where the two circle of mutual intersection becomes a independent circles overlap we see the sphere when we transfer the bottles to sphere of ambiguity. It has inscribed fourspace. This is because they may be within it two tetrahedrons in reciprocal oriented in any direction with respect to relation to each other (point for face). each other. If the singularity in the These two tetrahedrons embody the two mobius strip is the line of its edge, and if twists of the two Mobius strips that it is the circle of self-intersection in the make up the pentahedron in such a way Kleinian Bottle then it is certain to be that they overlap each other. three dimensional in the Hyper Kleinian Bottle. This sphere of mutual self/other What becomes clear as we explore this intersection is called the "sphere of strange geometry of the Hyper-Kleinian ambiguity." That is because at that point Bottle is that it is concerned with the of local incoherence it is impossible to interaction between three dimensional tell if the bottles are intersecting self or space and four dimensional space. Four other, and whether they compose one dimensional space is composed of four geometrical form or two, and whether three dimensional spaces in a quaternion there are one or two non-dual surfaces. relation with each other. Every three Each of the pentahedrons contains five dimensional space is a slice of four tetrahedrons. If we select any one of dimensional space. It produces two half those tetrahedrons then they may be four dimensional spaces (anna and

26 Deep Mathematics and Meta-systems Theory by Kent D. Palmer kappa) on either side of it. And antipodal be looked at from either anna or kappa to the three dimensional space is a single four dimensional subspaces. These axis that separates the two four subspaces can be thought of as mirrors dimensional half spaces at the other that are reflecting the sphere from either extreme away from the three the anna and kappa side. These sides are dimensional slicing plane. This means actually in all directions from three that there are twelve virtual axes in four dimensional space. This mirroring is dimensional space. These are mapped by such that the sphere looks the same from the quaternion twists into the four real either the anna or kappa side. There is axes of four dimensional space. We only no difference in the viewpoints. This is a see the virtual axes when we look at four very significant point. Viewpoints are dimensional space from the third unified non-dually in four dimensional dimension. If we use up three of the space with respect to the Hyper-Kleinian virtual axes in the three dimensional Bottle. Looking at the sphere of slice, and four in each of the four ambiguity from the point of view of the dimensional half-spaces on either side of Anna bottle is the same as looking at it the slice, then the final virtual axis from the point of view of the Kappa appears antipodal to the three bottle. We can make a transformation dimensional slice. As we will see shortly from one viewpoint to the other this is very significant. But if we look at seamlessly just as is suggested by the the Hyper Kleinian Bottle we see that its Lorenz transformation with respect to main thrust is the connection between inertial planes in Relativity Theory. the three dimensional slice and the four Mobius Strips make distinctions non- dimensional container-space. The two dual. Kleinian Bottles make the inside Kleinian bottles each inhabit one of the and outside non-dual and erase two four dimensional half-spaces. They distinctions. The Hyper-Kleinian Bottle intersect in the sphere of ambiguity in makes viewpoints non-dual and erases the three dimensional slice. If we inside and outside. In the hypersphere understand the Hyper-Bottle to be a any particular point may be on the three dimensional form involuted by inside, outside or on the surface of any mirror reflection then we can see the three dimensional sub-sphere of the sphere as having two Mobius strip 'ears' hypersphere. Inside and outside becomes which poke out into hyperspace. These ambiguous in four dimensional space. At ears reflect each other and return to the its great circle boundary these three sphere. Or we can see the Hyper-Bottle dimensional spheres shrink to points so as a bundle of spheres over a circle such that the hypersphere has a definitive that the spheres are mirrored though four boundary. But within that boundary and dimensional space but intersect at the with respect to three dimensional same circle. The dual Mobius strips can representations or sub-spheres there is be seen as the two sides of the non-dual no telling whether a point is on the self-intersection of the set of spheres that surface or on the inside or on the outside twist around each other in four of the hypersphere. The inside/outside dimensional space. All of these distinction becomes ambiguous except geometrical formulations are equal. But for the great circle outer surface of the the gist of their equality is that the hypersphere. When we lose our sphere in the three dimensional slice can distinction between inside and outside

27 Deep Mathematics and Meta-systems Theory by Kent D. Palmer we gain the non-dual fusion of Rosen calls this non-dual duality (or we viewpoints. All the viewpoints are may say chiasmicly dual non-duality). merely the vectors of four dimensional The tesseract is from one perspective mirroring. In this mirroring the coherent and from another incoherent. hypersphere looks like a bundle of So it is with the Hyper-Kleinian Bottle. spheres that get smaller and smaller till It is from one perspective dual and from they become points at the great circle another non-dual. It is dual in that it is boundaries. Within that boundary there composed of two different dual is a fractal arrangement of interfering mirroring Kleinian Bottles. But it is non- spheres that are mirrored in all directions dual in that those two Kleinian bottles to infinity. Each vector viewpoint from are conjuncted at their self-intersection the fourth dimension makes the fractal point so we cannot tell them from each sphere look identical. The same is true of other and so it becomes ambiguous pentahedrons inscribed in the whether they are one or two. The hypersphere. They form a fractal tesseract is super-rational while the arrangement that mirrors off in infinity Hyper-Kleinian Bottle is paradoxical in from what ever direction you look at it in this case. So we see that the set of four dimensional space. Each figures in the series Mobius, Klein, pentahedron has a dual in the orthogonal hyper-Klein interchanges places in the four dimensional half-space. Similarly fourth dimension with the series all the three-spheres of the hypersphere Penrose, Necker, Tesseract with respect have a dual that also appears to be to the embodiment of super-rationality mirrored to infinity getting smaller and and paradoxicality. Thus we see that smaller as it approaches the points on its paradoxicality is within super-rationality hyperspherical surface. and vice versa if we go deep enough.

This non-dual fusion of viewpoints in This complementarity extends to the four dimensional space is the structure of four dimensional space fundamental basis of the socius. Within itself. Four dimensional space is as we the social field there is a non-dual said four three dimensional spaces in a unification of all the individual quaternionic relation to each other. In subjective viewpoints prior to their the Hyper-Kleinian bottle we see the individuation. This is the underlying relation between the four dimensional basis of the social and is what allows us half-spaces either side of the three to have reflectivity. The four dimensional slice. The sphere of dimensional realm fragments the views ambiguity, localized incoherence, and produces 720 degree mirroring in all resides in the three dimensional slice. directions but at the same time makes That sphere looks the same from either the viewpoints non-dually fused, so that the anna or kappa side of the three there is a single non-dual viewpoint that dimensional subspace (hyperplane). All appears to be diverse in lower viewpoints on the sphere are non-dually dimensional slices which we have access fused in hyperspace. Viewpoints to. Thus in the fourth dimension themselves are like the dual mirrored coherence and incoherence become the Kleinian bottles that appear on the anna same thing. Similarly duality and non- and kappa side of the subspace. The duality become the same thing. Steve subspace takes up three of the twelve

28 Deep Mathematics and Meta-systems Theory by Kent D. Palmer virtual axes of fourspace that appear | when we consider it to be composed of |………..projective plane | (octonion) four three dimensional spaces. Eight 3 virtual axes are taken up by the two four (sphere of ambiguity) dimensional subspaces. That leaves one antipodal axis to the threespace. The it divides them us instead in this way: combination of this antipodal axis and 1 the threespace is a projective plane. The | projective plane is the dual of the Hyper- (complexnion) 2 | 2 (complexnion) Kleinian Bottle. They are | complementary. The Hyper-Kleinian |…asymmetrical octonion Bottle is ambiguously dual made up of 7 lower dimensional non-dualities. The When we look into these two patterns Projective Plane at the same level of what we see is very significant. One abstraction is non-dual due to its non- configuration of fourspace virtual axes orientability for lower dimensional defines the relation of the asymmetrical figures in threespace. The two four- quaternion to the symmetrical octionion dimensional half-spaces contain while the other defines the relation of the Kleinian bottles that define the sphere of asymmetrical octonion to the ambiguity. The sphere of ambiguity symmetrical quaternion. Here the defines a threespace slice out of imaginaries are the virtual axes of fourspace. The threespace slice is fourspace themselves. So the connected to the antipodal axis to create complementarity of duality and non- a projective plane that is non-dual at the duality brings us back to the quaternion four dimensional level without which represents the autopoietic and the ambiguity. The antipodal axis when octonion that represents the reflexive. taken with all the other antipodal axes of Reflexivity is embedded within the constituent threespace slices of autopoiesis and autopoiesis is embedded fourspace defines the unification of all in reflexivity. The asymmetrical possible viewpoints into a single non- configuration of the quaternion is related dual social viewing manifold. to the symmetrical octonion and vice

versa. In the latter figure seven When we realize that the structure of dimensional space is antipodal and fourspace is defined by the relation of produces an analog to the projective the projective plane and the hyper- plane. This is orthogonal to two Kleinian bottles then we recognize that dimensional spaces which embody the this is the dual of another complexnion. We may think of these complementary configuration of TWO two dimensional spaces as fourspace. That complementary articulations of the mandelbrot set. They configuration would divide up the virtual intersect to create a quaternion axes differently. Instead of dividing mandelbrot. The opposite of these is an them up so: octonion mandelbrot first pictured by (antipodal axis) Onar Aam based on a hypothesis of the 1 author. We further hypothesize that the (quaternion) | (quaternion) quaterbrot and the Aambrot (octobrot) (Kleinbottle) 4 | 4 half-(Kleinbottle) sets are the dual of the Hyper-Kleinian

29 Deep Mathematics and Meta-systems Theory by Kent D. Palmer bottle and the Projective Plane. They where all the projective planes intersect. form a Yin/Yang formation equivalent to It is the opposite of the sphere of the Great Ultimate in which on one hand ambiguity. It is non-local global we have infinite fractal sets that coherence of all viewpoints. The intertwine and on the other we have the seeming fragmentation of the socius as Hyper-Kleinian and Projective Plane that social field is based on this higher reconcile duality with non-duality and dimensional non-dual fusion of coherence with incoherence. viewpoints. This is what makes resonance between desiring and When we put these geometrical disseminating machines possible. excursions in the context of the Reflexive we see that a peculiar picture Reflexivity occurs within the arises of what Onar Aam hither to called mirrorhouse. All the paths of lightrays the Mirrorhouse. In the mirrorhouse of though the mirror house are non-dually the social there is infinite mutual fused into a single higher dimensional mirroring which underlies all social path via Projective Plane. All the interactions. We see this in the viewpoints within the Mirrorhouse are progression of mirror configurations non-dually fused together via the Hyper- from the mutual facing mirrors Kleinian Bottle. The views of two analogous to the complexnion and bottles on the same sphere of ambiguity embodies the underlying dissipative are the same. The reflexive space is structures, the three triangularly facing fragmented by the series of hyper- mirrors analogous to the quaternion mandelbrot sets despite its higher embodies the autopoietic structures, and dimensional unity. So the mirrorhouse is the inwardly mirroring tetrahedron that infinitely deep fractally and infinitely is analogous to the octonion embodies complex chaotically. Asymmetrical the reflexive structures. The reflexive quaternions are embedded in octonions level of mirroring produces an enclosed split into symmetrical quaternions. reflexive space. This is the analog for the Asymmetrical Octonions are embedded sphere of ambiguity. It is ambiguous in quaternions split into symmetrical space from the viewpoint of surrounding complexnions. This dual embedding flat space. But internally it is an enclosed structure gives us the complemetrarity mirroring surface. The space enclosed by between the autopoietic reflexive and the that mirroring surface has the nature of reflexive autopoietic. The autopoietic the Palmer Chiasmic Cube of self special system has a reflexiveness in the intersecting non-dual reversibilities. It is organization of its self-producing nodes. a non-dual field that borders on absolute The reflexive special system which is non-duality of interpenetration. In it intrinsically social is made up every viewpoint has an opposite that it autopoietic individuals. The interior of can be transformed into and all the autopoietic system mirrors viewpoints intersect at a single point that multidimensionally the exterior of that is a singularity which we call following system where multiple autopoietic M. Henry the Essence of Manifestation. systems resonate and interact. This is an The Hindu's called it Atman. Hegel image of the mirroring of called it Spirit. It is in the social non- interpenetration. dual viewpoint manifold as the point

30 Deep Mathematics and Meta-systems Theory by Kent D. Palmer

Paradoxicality and Super- Excluded Middle which is Aristotle's Rationality prime dictum. Indian Logic posits four statements: A, ~A, Both and Neither. In our unfolding of the roots of Nargarjuna shows us that Emptiness is Mathematics below the surface of the the difference between the Both and the Pure Presence Being we have uncovered Neither. Emptiness is super-rational. some anomalous mathematical entities That is why Koans may indicate that we have used as a basis for enlightenment as they specify the modeling the transition between Systems languaging that might occur at the point and Meta-systems. We recognize all the of entry into enlightenment, which is categories of mathematics as systems merely the realization of emptiness as and we see them in the field of all the fullness of interpenetration. When possible categories which is their meta- Steve Rosen and I discussed the system. Mathematics does not thematize topological series and its non-duality I the meta-system that encompasses the warned him that it expresses the super- categories. When we do thematize it we rational just as much as the paradoxical find that the transition into that meta- which he had emphasized in his works. system can be modeled by a series of He said that he did have in his though a emergent stages and those emergent place for what was not paradoxical stages can be seen mirrored in the series which was the Leminscate which is what of topological anomalies or the series of you get when you cut the mobius strip in hyper complex algebraic anomalies or half, i.e. a two sided band which is even in the series of paradoxical figures. twisted twice. This made me realize that So from this vantage point we envision there was a good chance that the functors that go between the different Leminscate represented the Super- series of anomalies that give us different Rational and that the other forms in the ways to talk about the same three topological series were devolutions from emergent levels between the system and that toward paradoxicality. The fact that the meta-system. the Hyper-Kleinian bottle defines the sphere of ambiguity enforced this But in some private discussions with realization. Steve Rosen on the Octonion Appreciation Society email list it Now what we realize is that actually the became clear that there was an important topological series is the reverse of the difference between the topological series algebraic series in terms of the definition and the algebraic series which can only of the movement from System to Meta- be seen when we take into consideration system. This means that the topological super-rationality and its relation to series has another level of paradoxicality. Super-rationality as we correspondence with the algebraic series have said appears in Koans of Zen which is the opposite of that outlined Buddhism. They are statements that take above. In this correspondence it is the us beyond what is rationally Hyper Kleinian bottle that is related to comprehensible. But Nagarjuna gives us the dissipative system and the mobius an even better way of thinking about the strip that is related to the Reflexive super-rational. He uses Indian Logic as system. We can realize how this is so if his basis. Indian Logic does not accept we think of the series of paradoxical

31 Deep Mathematics and Meta-systems Theory by Kent D. Palmer figures (Penrose, Nekker and Tesseract) reflexive system becomes the sphere of as the outward appearances and that the ambiguity. At the super-rational level topological series gives us the inward there is a clear distinction between the realities associated with these two autopoietic systems which is at the appearances. Thus the Hyper Kleinian same time their own distinctions and the Bottle defines the implicit reflexivity of other’s distinction. In other words at the the dissipative. This is to say that the super-rational level distinctions are structure of the dissipative system mutual resonances. Between these two combines a circuit of Logos with a ends of the spectrum the distinction goes circuit of Physus into a single loop that through a series of devolutions from the appears outwardly to act like the Escher superrational through the mobius strip waterfall. But this outward difference and the kleinian bottle and the hyper- between the loop of logos and the loop kleinian bottle. At the first stage the of physus becomes one though the inner clear distinction becomes non-dual in the possibility of the Hyper-Kleinian Bottle. mobius strip, i.e. locally dual but It is that which makes the singularity and globally non-dual. Then it devolves the boundary of the dissipative system further to the point where the distinction one yet not one at the same time. As the is lost in the Kleinian Bottle but the reflexive properties become externalized inward and outward becomes non-dual. then we see the over-spilling of two Finally the circles of ambiguity of each dimensional paradoxical appearances kleinian bottle, i.e. each autopoietic into the four dimensional tesseract. The system, becomes intertwined. This inward possibilities become simpler so produces the hyper-kleinian bottle that we see within the reflexive system configuration where the self cannot only the non-duality of the mobius strip distinguish itself from the other. Finally that is one step away from complete this degenerates into the sphere of interpenetration in the Meta-system. ambiguity where there is perfect paradoxicality. If the sphere of If we realize that the two series are ambiguity itself involutes then it reversed with respect to each other from becomes meta-paradoxical or absurd. a certain point of view then it becomes clear that the topological series is telling This picture of the autopoietic system us something different from the producing its distinction with respect to algebraic series. When we consider what another autopoietic system in the that might be then we realize that if we reflexive environment and the take a pair of autopoietic systems in a devolution of these distinctions in a reflexive milieu that they must quantal series gives us an even more continually define their own and each precise model of the special systems in other’s boundaries. This self other their relation to each other. The definition that occurs within the difference between the topological series reflexive system is governed by the and the algebraic series is an important topological series. In other words it discovery that has far reaching becomes clear that the definition of the implications for the modeling of boundary may range on the spectrum autopoietic social systems. from super-rational to paradoxical. At the level of the paradoxical the whole Wavicles and the Emergent Meta-

32 Deep Mathematics and Meta-systems Theory by Kent D. Palmer system instantatons that pop around to create their statistical shell around the atom. In Solitons are solitary waves that act like this scenario electrons do not actually particles. We might call them wavicles. move. Instead they vanish one place in These non-dual waves can be seen to their electron shells and then play an important role in the definition momentarily appear somewhere else in of dissipative, autopoietic and reflexive the shell. This process repeated over and systems. over gives us the image of an electron moving around its shell. Besides the topological series, the algebraic series and the series of I combine this idea of the instantaton paradoxes there is another important form of the electron to create the idea of anomalous series that must be the Super-breather. The super breather is considered. That is the series of Soliton, composed of two breathers separated in Breather and Super-Breather. Solitons spacetime exchanging solitons via have long been posited to be the model instanation jumps. Such a formation of super efficiency that gives us a would give us just the kind of properties physical representation of the dissipative that we would expect at the reflexive system. It is known that Soliton waves level. It would give us information and form Breathers when a Soliton and an energy exchange effortlessly at a Anti-soliton (hole) interact. This causes distance between autopoietic nodes. a stable structure where the soliton falls Thus if each autopoietic node is a continually into the anti-soliton and the breather formation then the super anti-soliton falls into the soliton. This breather is a means for each node to gives us a ‘breathing’ motion that is participate with the other nodes in the static in space. We have long thought of network exchanging information and this as a analogy for the autopoietic energy across the intervening spacetime special system. Also we have had the within the autopoietic system. A similar hypothesis that there is a super-breather kind of exchange may be taking place that is the conjunction of two breathers between autopoietic systems within a that corresponds to the reflexive special reflexive system. This ability to system. This hypothesis was exchange information and energy at a strengthened when the form of the distance between breathers can explain Hyper-Kleinian bottle was recently how the autopoietic system can be so discovered by the author. This led to the stable and energy efficient. It can also postulate of the form of the Super- explain how it can achieve its self Breather. Solitons have another form identity as a basis for self production. which is called the Instantaton. The What Jahn and Dunne call quantum instantaton is a soliton that pops from tunneling between consciousnesses may point to point by moving though troughs be exactly this kind of super-breather in the potentials of a field. They seem to sort of information and energy exchange. move across spacetime instantaneously We know that the nodes within the without passing through the intervening autopoietic system must be an image of space. They merely pop from one place the whole of the autopoietic system to another. There is a theory that itself. We think of these nodes as a kind electrons and other particles are really of dynamic hologram where each part

33 Deep Mathematics and Meta-systems Theory by Kent D. Palmer mirrors all the others and this is how autopoietic nodes as being produced by self-identity is achieved. But the super- continual creation and destruction in breather concept allows us to understand very small time quanta. the dynamic resonances that underlie self-identity of the autopoietic system. The seeds of the autopoietic system are the memories of the prior period. We Here again we can see how another access those memories and we produce a anomalous series can give us new picture or vision of the self of the information about the special systems. In autopoietic system. The nodes then this case we are learning not about the attempt to reproduce this vision or distinguishing of one autopoietic picture. They each have their own systems from another but about the viewpoints on the rest of the swarm of dynamics of action at a distance where autopoietic nodes. They produce there are exchanges of information and candidate lists based on their mutual energy in quanta across spacetime. This action to survive to the next cycle. Then ability explains how the autopoietic the candidate lists annihilate until there nodes exchange information necessary to is only the seeds for the next cycle and establish their self identity although they those seeds become the memories from are all in different places. It also explains which the new EMS cycle will arise. what the quantum tunneling between consciousnesses described by Jahn and The EMS cycle is produced out of the Dunne1 might consist of at the social special systems. Each special system is a level. meta-operator within the Emergent Meta-system. The nodes of the autopoietic system may be seen as a swarm of monads that Real System = creation appears in the Emergent Meta-system Dissipative System = annihilation (EMS). Super-breather instantaton hops Autopoietic System = mutual action might be considered the means of Reflexive System = gestalt pattern moving from one EMS phase to another. formation Thus we might consider that the nodes Meta-System = Emergent Meta-system of the autopoietic system move though an EMS cycle from seed to monad to These form a cycle: viewpoint to candidate as the means of MONAD VIEWPOINT establishing its self-production. Mutual Mutual Action action between the autopoietic nodes Gestalt may be via the instantatoin super- Creation Pattern Formation breather soliton exchange. Gestalt pattern formation may be though the Annihilation SEED CANDIDATE process of establishing distinctions governed by the topological series. This cycle solves the problem of how we Annihilation and creation occur when can have radical emergence without the autopoietic system pops into and out causation. This is a fundamental problem of (apoptosis) existence. Or alternatively posed by Mahayana Buddhist we can see the illusory continuity of the metaphysics. The EMS cycle solves this problem by positing that there is an light 1 Jahn & Dunne Margins of Reality

34 Deep Mathematics and Meta-systems Theory by Kent D. Palmer bouncing off a inwardly mirroring  Real tetrahedral surface. Conway showed that  True there is one such path that allows a ray  Identical to return to its origin within a  Present tetrahedron. The EMS cycle follows this path and bounces off the mirrored It has been shown by Butchvarov2 that surface producing a cycle in which there material identity implicitly assumes that is a transformation each phase of the there are pre-entities that are identified. cycle that moves though the series from We can see from this that each kind of the System though the special systems to pattern has an inherent relation to a the meta-system. In this way the meta- particular aspect of Being: system continuously reproduces itself non-causally across time. This model is Value = Real similar to Ben Goertzel’s Self- Structure = Identity Generating “Magician” System model Sign = Truth except it allows for creation ex nihilo. Process = Presence We can construct the same model with instantaton jumps between breathers in a This structural ontological model that super breather if we consider each posits pre-entities gives us a basis for autopoietic node within an autopoietic projecting the emergent level of pattern system to be a breather. Breathers are below that of form. Patterns are the continually annihilating and creating content of forms and this content comes themselves as the positive soliton falls in four types. Klir has described process into the negative soliton and vice versa. and structural kinds of pattern in The super-breather formation allows this Architecture of Systems Problem to occur across spacetime in such a way Solving. Baudrillard has described the that information and energy is sign and value kinds of pattern in exchanged. This exchange may be seen Critique of the Economy of the Sign. All as an image of the Emergent Meta- four of these may be seen to follow the system formation. pattern of the epistemological hierarchy posited by Klir. They all give rise to Disipative Autopoietic Reflexive meta-levels of patterning and also the System Modeling different kinds of pattern entwine chiasmically. In this section we will use the information from the different series of What we like about this formation is that anomalies to construct a model of the it makes possible a basis for describing Autopoietic System in a reflexive the autopoietic system. That system has environment. This model makes use of a a control hyper-cycle that exists in an new realization that there is an inherent imaginary “space” over and above the relation between the minimal method spacetime in which the autopoietic nodes viewpoints, the aspects of Being and the appear. The kinds of pattern allow us to kinds of patterns. describe that imaginary “space” in terms of sign/value as opposed to the real The aspects of Being are as follows: 2 Butchvarov, Panayot Being Qua Being Indiana University Press

35 Deep Mathematics and Meta-systems Theory by Kent D. Palmer space produced by process/structure. real-time system which has been Process is the meta-models that dictate previously identified in Wild Software when structural changes in our systems Meta-systems by the author. model must change. Structures dictate the patterning of values of variables.  Process Sign = event They are like functions over those  Process Value = function variables. The Process meta-models  Structure Sign = agent dictate the circumstances when the  Structure Value = data functions or the linkage between functions change for a given variable. Once we have realized this connection Together they give a model of the then everything that is known about the spacetime changes in a system. But sign four methodological viewpoints comes and value may be seen as nowheres that into play within the autopoietic system. are the basis of producing process or We may posit that the boundary between structure based dissipative systems. Sign the imaginary realm of sign-value and is a no-when and Value is a no-where. the spacetime realm is like awareness. Sign and process create a dissipative What is controlled by the hypercycle structure and Value and Structure create exhibits intentionality and thus is a dissipative structure. When these two directed awareness or consciousness. dissipative structures come together they What is not controlled are merely create an autopoietic system. In that statistical processes and these are autopoietic system sign and value may unconscious. This also helps us define be seen to define an imaginary space the difference between organization and over the spacetime of the process and what Maturana and Varela call structure. structure pair. In that imaginary space Organization is what is intentionally sign-value pair vectors produce a controlled by the hypercycle while network within which hypercycles can structure is the patterning that goes on in form. These hypercycles are the control the spacetime realm that is not controlled structures for the autopoietic nodes explicitly in the process of self within the autopoietic system. production. This also allows us to relate this model to Jungian psychology which Imaginary Realm defines the self as the totality of the Sign-Value conscious and the unconscious. The ego network is the limited identity that is totally self- hypercycles conscious. The ego exists in the Logos of the imaginary realm. The self includes node node the spacetime embodiment as well as the ego. The self is the timespan it takes the Real SpaceTime ego to become itself. It is the quantum of Process Structure time necessary for it to produce itself

again out of itself. Awareness is the non- Now here is an interesting point. Once dual man between the conscious and the sign-value hypercycles form then the unconscious. Processes and Structures can relate to either Value or Sign independently. This The four viewpoints give rise to the produces the four viewpoints on any sixteen minimal methods. The

36 Deep Mathematics and Meta-systems Theory by Kent D. Palmer viewpoints appear right at the boundary EMS lifecycle. This is done by each of awareness. Thus the sign-value monad producing an internal model of networks may be seen in terms of petri its relations with the other monads based net and state machine control structures. on its experience in the mutual action We note that for the monads there must phase. Those models are completely be a transformation into viewpoints in internal and there is no relation between the EMS cycle. We posit that the arising the separate models that are produced. of these relations between the kinds of However we can have a fuzzy summary patterns are the means of by which the externally of the different posited monads archive their viewpoints on relations. The monadic autopoietic nodes themselves. have the form of computational monads. Computational monads are constructed Given this insight it should be possible out of the aspects of the minimal to reconstruct the actual EMS cycle that methods that do not require any relation the autopoietic nodes participate in. We between design elements externally. We note that there is a sharp distinction are thus moving from Laws of Form to between the sign-value networks and the Laws of Pattern as the monads develop a process-structure spacetime in which the layered model of their posited relations autopoietic nodes exist. It might be with other monads in the swarm of possible to see the seeds of the EMS autopoietic nodes. Out of their separate cycle as the sign-value pairs in the models the monads posit what other hypercycle networks and the activation elements should be in their solipsistic and deactivation of nodes as the creation design. These elements are drawn from a and annihilation meta-operators. Now if structural basis in which virtual monads this works then it is possible to see the and virtual anti-monads are produced. hypercycles as the seeds and the Different monads select from this autopoietic nodes and the monads. The structural virtual soup those it would like seeds activate and deactivate the nodes to include in its design. The design may and that corresponds to their creation take the form of a non-well-founded and destruction. The nodes form a hyperlist (instead of hyperset). This is swarm of monads. These monads engage kind of like the different buildings that in mutual action when they are activated. MUD users create independently of each Out of that mutual action comes the other. Each user creates the rooms and arising of viewpoints. The viewpoints furniture he or she wants in their virtual arise though the interaction between the environment adding on to the already sign-value and process-structure arenas. built maze that others have created. Mud Process can relate either to sign or value users can destroy the work of others by as can Structure. The sign-value not allowing them to connect their networks get interpreted as petri nets or buildings to the already existing state machines. The monads are viewed building. Each user decides what it will in terms of their autonomy as agents, allow to be connected to his own their functionality as transformers, their buildings. Thus the rhizome of the data and events which intertwine in the buildings in the MUD is controlled by self-production process. Self production each independent user allowing or is seen as the production of candidates disallowing connections. Now we can for propagation to the next cycle of the see a similar structure in the internal

37 Deep Mathematics and Meta-systems Theory by Kent D. Palmer design of the computational monads that sees as important. These architectures are the autopoietic nodes. Each monad may be non-well-founded hyperlists creates the design of the swarm using the where monads can be members of their minimal methods as it sees fit. Each uses own hierarchies and where set members structures of monads that are drawn from may be repeated many times in different the structural pool which are opposites. contexts. Thus each monad develops a When we take these various structures vision of the whole swarm of autopoietic produced by each monad as a hyperlist nodes. They can communicate about and place them together then the their visions and share information elements and anti-elements cancel each though mutual action. This mutual action other out. This leaves the seeds for the as we have seen is probably carried next life cycle phase. Now the question though super-breather exchanges of is how this EMS cycle can be the basis instantons at a distance. Externally we of self production. can only see fuzzy summaries of the overall structure of the swarm Self-production occurs when there is architecture. What is interesting about interaction between the autopoietic this is that it is possible to apply genetic nodes that are embedded in process and algorithms to the production of the structural patterning and the network of architecture. We merely produce swarm sign-value pairs that make up the members randomly which have different hypercycles. In this interaction the first internal models. The fitness functions moment is the activation of certain are applied to the fuzzy summary of the autpoietic nodes and the deactivation of architecture which then applies others. This inhibition and excitation by evolutionary pressure to the individual the hypercycles produces a kind of monads. Now this fitness function cancellation because inhibition and allows us to select which monads will be excitation by different hypercycles excluded from the swarm based on the cancel each other. Given a set of overall fitness of the swarm and the autopoietic nodes that are turned on then individual monads contribution to that they develop viewpoints from the fitness. Thus we can imagine two interaction of process and structure different forces acting in self production. separately with either sign or value. This Externally we can see genetic algorithms produces the agent, function, event and and internally the cancellation of the data viewpoints. Out of the interaction of individual designs of the monads. Self these viewpoints arise the minimal production is a tension between these methods. The sign-value pairs get two different factors. Self-production is interpreted as petri-nets or state seen as the evolution of the swarm machines and these allow control to where the self is the essence of the occur in a fashion like software species that is produced by crossover programs. All the minimal methods may and mutation. Cross over occurs when be employed first to define the nodes as hyperlists are shared. Mutation occurs computational monads and then to build when a new monad is produced with a up inside the layering of each one an randomly generated hyperlist of swarm image of the design of the swarm where design. But internally Self-production is each node posits the architectural the collusion between nodes that allow connections to all the other nodes that it particular nodes to survive from one

38 Deep Mathematics and Meta-systems Theory by Kent D. Palmer lifecycle to the next. Ben Goertzel crone and an archetype of the Self as a describes this collusion as the social totality. The shadow is the anti-swarm to basis of Self-Generating Systems. Self any specific swarm signature lifted out Generating Systems assume of the structural soup of nodes and anti- discontinuity rather than continuity as nodes. The animus-anima is the the basis of the meta-system. It is this mirroring of the anti-node in the node assumption that makes the Autopoietic and the node in the anti-node. The wise System a partial meta-system or a old man/chthonic female crone is the broken and fragmented system rather next meta-level mirroring of node-anti- than a whole with continuity. Yet the node or anti-node-anti. Finally the Autopoietic partial meta-system can still archetype of the Self is the EMS produce itself. It does this by the structure itself. Each of these mirrorings collusion of the Autopoietic nodes in the take us up one level in the hierarchy of EMS cycle. Each one persuades the Hyper-complex algebras. Each mirroring other not to cancel it out by including it shows us the depth of time where the in their hyperlists rather than the its nodes look into their future or past and structural opposite. We see here that see that in the generations of other nodes structuralism is the basis for that surround them. Like the forest that cancellation. Process on the other hand has trees of all ages the nodes can look is what changes out structures over time. at other nodes that are at earlier and later Process is what produces the stages of their lifecycle within the EMS segmentation in time of the EMS lifecycle. Self Production must have a lifecycle that is occurring inside the genetic development structure over time autopoietic system. Signs are the basis of governed by Process. What the self is all reference within the Autopoietic that is being produced is continually system and also self reference. It takes changing as the essence of the individual two sign-value pairs to point back at organism unfolds in their genetic itself and produce a minimal hypercycle development. Self production is the which merely cycles back into itself production not of the identical but of the endlessly. Value is what allows the Same. This sameness can be produced hypercycles to set minimums and because of the Belonging Together of maximums on a particular parameter and the autopoietic nodes in the swarm. thus allow control of that parameter. Their belonging together inwardly is a There must be a network of parameters collusion in which the group allows the that represent the organization of the individual to survive from cycle to cycle. Autopoietic system. This network is But outwardly the belonging together is used as an envelope of constraints that their mutual filling of a niche in the governs when certain nodes are turned environment and their adaptation to that off and on. So self production occurs in niche by the response to selective terms of external fitness and internal pressures. cancellation and collusion. By self we mean the totality of the self including This concept of the autopoietic nodes as awareness, intentional consciousness and an EMS cycle is a very sophisticated the unconscious. Each autopoietic vision of the Autopoietic System. We system has as Jung says a shadow, an already know that the Autopoietic nodes animus and anima, a wise old man and are themselves autopoietic systems.

39 Deep Mathematics and Meta-systems Theory by Kent D. Palmer

They have a quaternion formation that that eventually participate externally in a makes them like mirrors that reflect the genetic algorithm and internally in the rest of the swarm. The swarm of cancellation process that allows for autopoietic nodes has the same relation collusion. to each other as the set of autopoietic systems have within the reflexive social In order to specify more precisely what environment. There is a social we have said generally let us proceed to environment inside the Autopoietic talk about the steps of the arising of the system that reflects the Social Emergent Meta-system as the means of environment outside it. This refection Self-production in the Autopoietic between inside and outside makes the system. First the general vision is that Autopoietic System istransparent even we have at least two autopoietic systems though it is informationally closed while within a reflexive environment. The energetically open. So we can see that autopoietic systems have forms the autopoietic system is awash in a governed by the quaternion algebras and reflexive social sea but it also has that the reflexive field is governed by the reflexive sea inside itself. Now we octonion algebras. Within the realize that the autopoietic system and autopoietic system we follow Maturana the autopoeitic nodes that make it up and Varela by seeing a set of autopoietic BOTH achieve their self production by nodes that together somehow manage to participating in an Emergent Meta- produce the autopoietic system by system cycle. There are two EMS cycles working together, i.e. socially. Now going on both on the Macro and Micro what we are positing is that these nodes scales at the same time. By the micro are members of an swarm and that they cycle the Autopoietic system produces self produce in the form of a Self itself. By the macro cycle the Society of Generating System (or Emergent Meta- Autopoietic systems produces itself. system). This means that the form of the These two processes mirror each other meta-system produced by all the meta- so that we can as Socrates suggests in operators that come from the entire set the Republic look into the soul of the of special systems work together to individual by looking at the way that the create the self-identity of the autopoietic city works. Self Generating meta- system over time. This is a radical systems or Emergent Meta-systems as I concept of how self identity is produced call them produce the self as a totality as because it is based on the assumption of Jung suggests and are not merely the radical discontinuity instead of perpetuation of the self conscious and continuity which is the usual foundation self identical ego. This is possible of systems theory. We are saying that the because all the aspects of Being (reality, Autopoietic Special system is really a identity, presence, and truth) interact partial meta-system and a fragmented chiasmicly though the kinds of patterns system and thus falls between the that exist at the pre-entity level. This System and Meta-system. Thus, we interaction produces the viewpoints cannot assume continuity as we would in (agent, function, event, data) which in a normal systems theory. Therefore, we turn allow the minimal methods to arise must assume discontinuity and explain which then form the basis for the continuities rather than the other way internal designs of the autopoietic nodes around. Self-production is no-longer a

40 Deep Mathematics and Meta-systems Theory by Kent D. Palmer creation of identity, but instead they are forms that are both nouns and something more like the Sameness that verbs at the same time. An autopoietic Heidegger defines as Belonging form is one where form forms itself. We Together in Identity and Difference. think of this process of form forming itself on the model of the inter- Now how do we approach understanding transformations of the quaternion. This the Autopoietic nodes though the means that we need imaginary marks. Emergent Meta-system formation as a We produce these imaginary marks by means of self production? We begin by rotating the normal marks used in saying that the autopoietic special normal Laws of Form system is caught in Being as an eventity. It thus has the same aspects that Being has answering to truth, reality, identity and presence. Truth and Identity reflect Real the Logos and Reality and Presence mark k reflect Physus. So the autopoietic system has an appropriation to each of these aspects of Being. This shows up when i we consider the level of pattern. The j level of pattern can be identified to be composed of pre-entities that are These marks transform into each other appropriate to the four aspects of Being. according to the algebra of quaternions. These pre-entities are of four kinds: Due to this self-identity has a very process, sign, structure and value. Sign precise model. There is also a precise is related to reference, Value is related to interface between the imaginary realm of assignment of a signature out of a marks and the real marks. Given the typeset, process is related to the laws of form we can see that this allows changing of structures over time, and us to model three of the four kinds of structure is related to the actual bit level Being. Pure Presence Being is modeled coding below the typing super-structure. by the vertical staves of the mark. Each of these kinds of pattern defines a Process Being is modeled by the kind of pre-entity related to the aspects horizontal roofs of the marks. Hyper of Being. Before we can talk about Being is modeled by the jumps autopoietic systems we must first talk backward and forward from different about autopoietic forms that exist as an points in the nested marking structures. emergent ontological level between What is missing is Wild Being which pattern and system. At the level of form relates to the contents that the marks we want to consider G. Spencer-Brown’s surround. We now know that these laws of form as a reference formalism as contents considered as pure non- Varela does in his work on dynamic manifesting Hyle appear in four kinds. forms with Kauffman. We see the kinds We want a simple way to articulate these of pattern as the contents that fills these kinds of patterning of contents. We do forms produced out of ‘marks’. Because that by introducing the formalism of the G. Spencer Brown’s marks are both Surreal Numbers. Each kind of pattern is operators and operands we consider represented by a different style of up or them autopoietic forms. That is to say down arrow. The concatenation of the

41 Deep Mathematics and Meta-systems Theory by Kent D. Palmer arrows of a particular style pattern pattern and laws of form at will in the produces a surreal number. The surreal nesting of the two sets of laws numbers can fill the marks and give us a complete model of the kinds of Being ((/\ /\ { (\/ \/ /\ /\ ) ( /\ ) } \/ \/ ) /\ /\ ) which also represents all the aspects of Being at the same time. What comes out of the realization of the duality between Laws of Pattern and (/\ /\ /\ /\ \/ \/ /\ \/ \/ \/ /\)((\/ /\ /\ \/) /\ /\ ) Laws of Form is the fact that there are really four elements that both kinds of This is a pattern of surreal arrows in a Laws deals with namely: laws of form mark sequence. In order to fill this out we would have to consider  Form four kinds of arrows that relate to the  Nothing kinds of pattern (sign, value, process,  Layering and structure) and four kinds of marks  Multiplicity that represent the three sorts of imaginary marks. This gives us an Laws of Form and Pattern give different unified model of Being in terms of kinds priorities to either layering or and aspects. multiplicity. However, there is a fundamental misconception of Spencer- But there is another consideration. Laws Brown that needs to be cleared up. He of form is contrasted to what I have identifies the null with the void or previously called Laws of pattern that emptiness from Eastern religious has opposite laws: philosophy. This is wrong. The null is just an empty background. We only get Laws of Form = RULE to the void or emptiness if we posit an ()() = () anti-form. When we posit an anti-form (()) = null then we can say that something is both the form and the anti-form or we can say Laws of Pattern = ANTI-RULE that something is neither the form nor the anti-form. Emptiness or void is ()()= null between this both and neither. Thus we (())=() must have the possibility of including para-complete and para-consistent logic Laws of Pattern are not considered by G. in order to approach emptiness. In order Spencer-Brown. They are a model of to accomplish the we will introduce immanence instead of transcendence. Matrix Logic of August Stern. This logic They are a model that affirms layering allows truth vectors that include both instead of multiplicity. Thus we need to and neither truth values. However, we be able to switch back and forth between cannot apply these just to truth but must laws of form and pattern. It has been apply these to all four aspects of Being. suggested that we do that by introducing So we will have in addition to truth braces and saying that what ever lies vectors also reality vectors, presence outside the braces are laws of form. vectors, and identity vectors. It is though However the layers of braces allow us to this means of introducing truth, reality, pass back and forth between laws of identity and presence vector subscripts

42 Deep Mathematics and Meta-systems Theory by Kent D. Palmer and treating these with Matrix Logic that completeness. We introduce brackets to we introduce the level of System into be able to label sets with the same aspect our consideration. We consider that vectors. We get autopoietic organization Matrix Logic gives us the organizational at the level of the autopoietic system the level of specification of the autopoietic way Stern says we do by the operation of system. Stern talks about the autopoietic matrices on each other in rings. At the rings when his rings of matrix operators level of pattern there are three different work on themselves. kinds of arrows that differentiate the kinds of patterns that the Surreal When we introduce the anti-form it is by Numbers are representing. structurally reversing all the properties of the form. In terms of Laws of This is a complex model. What can we Form/Pattern it is the reversal of all the say we archive with its different relations in a particular set of marks. elements? First of all we have a model with specific formalisms that address Form ( \/ \/ ) ( /\ ) (( \/ ) /\ /\ ) what autopoietic unity is at each level. Anti-Form ( \/ \/ ( /\ )) ( \/ ) ( \/ \/ ) At the level of autopoietic form it is the ‘mark’ of G. Spencer Brown from his The specific elements may have Laws of Form. At the level of different truth, identity, reality, and Organization we have the Autopoiesis of presence vectors associated with them: rings of matrix operators that August Stern provides in Matrix Logic. It allows R= r,r us to see how the features of the I= i,i autopoietic system can be built up from T= t,t the structural level of pre-entities by the P= p,p coming together of the different kinds of patterning. It explains how content and The palaces may have 1, 0 or -1 (hidden) form interact in terms of the containment in either place in the vector. Since we of the surreal numbers by the marks. It want to be able to mark Being aspect explains how autopoietic form appears vectors where ever we want in a series of in the production of quaternion marks we must introduce brackets to imaginary marks at the formal level. It group things with the same vector explains how form and anti-form appear designations. All this gives us the ability and how that is transformed into pattern to introduce very complex equations that and anti-pattern when we switch to the describe a system on the three levels of anti-rule set by reversing the laws of System, Form and Pattern. At the level form to produced the laws of pattern. of pattern we have Surreal Numbers. At Here pattern means something different the level of form we have both Laws of from the patternings at the structural Form and Laws of Pattern differentiated level that are differentiated into kinds. by Braces. We have a quaternion of Laws of Pattern refers to the emphasis of rotated marks. And we have the layering over multiplicity. Instead we possibility of expressing forms and anti- might talk of the rule and the anti-rules forms. At the level of System we have that occur when we reverse the rules that the vector operators that allow us to apply to the algebra of the marks. We express para-consistency and para- call this pattern because the emphasis on

43 Deep Mathematics and Meta-systems Theory by Kent D. Palmer layering rather than multiplicity focuses might be seen as the process of the anti- our attention on inwardness and soliton falling into the soliton and vice immanence rather than externality and versa. It produces a complex dynamic transcendence. This tends to make us space in which forms and anti-forms look at patterns rather than forms. cancel out but perhaps complements and However the "Laws of Pattern" as an anti-complements reinforce each other anti rule set is of a different nature than such that soliton and anti-soliton the patterning at the level of the pre- dynamics in the breather might be entities. We should not allow the use of modeled. Laws of Form never addresses the same word ‘pattern’ in two senses the question as to when expansion or confuse us here. condensation should be used. It merely identifies forms that are identical under a If we talk in terms of rules and anti-rules series of expansions and contraction then we clearly have Form and Anti- operations. We have instead a notion form under both regimes of rules. These that the autopoietic node is a super forms are composed of marks and breather. The expansion and surrealistic arrows as contents. If we condensation is the breathing. That reverse the order of the marks and breathing contains within it two solitons reverse the arrows we produce the anti that represent the two conjuncted form under a particular rule set. In laws dissipative systems which compose the of form there are two directions to our autopoietic system. In this case the application of the rules. We can process dissipation is occurring at the pattern them forward or backward depending if level, so that one is a sign-process pair we want to elaborate the marks or and the other is a value-structure pair. simplify them. If we apply the marks to When these come together we get the condense or elaborate it is possible to production of the hyper cycle network of elaborate using rules and condense using sign-value pairs on the other hand and anti-rules or vice versa. Forms and Anti- on the other we get the process-structure forms are assumed to cancel. But pairs that define the autopoietic nodes. It elaboration under rules and condensation is the activation of nodes by the hyper- under anti-rules may not lead to the cycles that produces the autopoietic same results. If we consider these system’s operation. The autopoietic elaborations and condensations of marks system is seeking homeostasis guided by under the different rulesets then we can the hyper-cycles. That homeostasis is a imagine that these are processes like the self-production which is ever different breathing of the breather soliton and from itself, i.e. includes the genetic anti-soliton pair operating on forms and unfolding of the organism over a anti-forms. We start with a form, we lifetime, just as the species unfolds in a expand under one rule and then at some different way over the generations. So threshold we contract under the other the mechanism that allows self anti-rules to get the complement form. If production is the core of the self- we do the same using the anti-form we generating system that operates on the the anti-complement form. The cycle nodes as if they were what Goertzel calls from form to complement form and back Magician Systems. Such systems act as to form is the dual of the cycle of the magicians in the sense that they all anti-form to the anti-complement form create each other in potential and then

44 Deep Mathematics and Meta-systems Theory by Kent D. Palmer their mutual potential creations cancel to give us the elements that exist in the next Rule:Form Anti-Rule:Form lifecycle. Self production is taken to be A Non-Anti-A ‘self’ in the sense that Jung uses the Rule:Anti-Form Anti-Rule:Anti-Form term, as the totality of the conscious and Anti-A Non-Anti-A the unconscious. The unconscious is seen as breaks in consciousness. Because there are two sub-rules for each Discontinuity is assumed and continuity rule there are many combinations of is explained. Thus we are really these for expansion or contraction. modeling a meta-system not a continuous system. The relations between rules and anti- rules will be further treated in the When we look at the breathing then the Formalism Specification Appendix. The sign-value soliton falls into the value- key point is that we are able to express process soliton and vice versa producing all the levels that are covered by the an energy cycle that is very efficient. It theory of autopoiesis by means of these is basically a self-producing trough interconnected formalisms. where the trough for each soliton is the other soliton. This breather is static in Autopoietic Organization = 4 aspects3 spacetime instead of moving though an Autopoietic Forms = 4 forms4 externally produced trough. It is an Autopoietic Structure = 4 patternings5 excellent picture of the cycle of self- production. When we try to model that However, our analysis reveals that using the laws of form-pattern and the autopoietic theory does not actually patterning from the pre-entity level then reach the level of system without the we get a complex landscape where deus ex machina of observers introduced expansion and contraction, form and from nowhere. In other words anti-form, rules and anti-rules intersect Autopoietic Theory as it stands is really to produce a forms and complementarity about autopoietic forms and Structure forms or anti-forms and complementary and Organization are used to define the anti-forms. This gives us something formal level but does not achieve the similar to the model of the Greimas emergent level of the system except by square that we discussed earlier. In that the sophistry of the sudden introduction model there is the form and the anti- of observers from nowhere. Reflexive form as contrast to the non-form. What Autopoietic Theory is more natural in we are saying here is that the non-form the sense that it merely defines system exists where the anti-rules apply instead and meta-system and then reveals that at of the rule. Thus the anti-non-form is the the point of balance between them exist complement to a particular form that we get by expanding using the rule and 3 Designation as Real, Present, Identical, True contract using the anti-rule. This tells us using Matrix Logic Vectors as superscripts on that there is a chiasm between one bracketed expressions. These aspects of Being possible anti-rule anti-form and another have a quaternion relation to each other. 4 in this complex landscape of possible The mark and the three imaginary marks associated with the i, j, k of the quaternion. form/anti-form or rule/anti-rule 5 The kinds of patterning that are associated with transformations. the aspects of Being are sign, value, process and structure.

45 Deep Mathematics and Meta-systems Theory by Kent D. Palmer the anomalous special systems. But we of the mark to include a dynamic as would still like to move up to the level being described by rules instead of of the system and understand that functions. The hyperjump that Spencer- emergent level of phenomena in a Brown describes now becomes the natural way with a precise formalism. means for iteration to occur and thus How that might be done will be allows us to approximate a rudimentary addressed in the next section of this programming language. paper. The function or rule marks may be Autopoietic Systems combined in such a way to produce ‘objects’ by realizing that the marks as When we attempt to move up to the data are different from the marks as autopoietic systems ontological level we functions and by making the provision need to formulate what is necessary in for segregating the data and operating on terms of our understanding of it via dedicated functions. Objects are Autopoietic Forms. These forms are combinations of forms in space (data) represented by Spencer-Brown’s marks. and forms in time (functions). Events We need to find a way to understand the can be seen as generated by the instances Forms of marks from a systematic of the firings of the functions and Agents perspective. We do this by extending can be seen as the autonomy of rulesets again the concept of the mark to cover which will allow simultaneous the difference between the mark in time execution. A generalization of the and the mark in space. The mark in time Turing machine created by Gurevich can be thought of as a function while the called the Abstract State Machine mark in space may be thought of as the Method allows us to describe arbitrarily data that is operated on by the function. complex systems by merely concatenating rules together. I have Function or Rule shown in a paper on this method that Rules have the special property of fusing ( /\/\/\) ((/\/\\//\)\/) f ((/\/\\/\//\)/\/\) all four methodological viewpoints into a single construct. A rule contains event, Inputs Outputs agent, function and data aspects in a single elegant construct. It also contains Hyper-jump the different layers of information, i.e. data, information, knowledge, wisdom. Data is the values that are input and When we make this addition to the output by the rule. Information is the marks we are basically producing a specific relations between data input and functional programming language out of data output. Knowledge is the them. We are giving new meaning to transformation that is made explicit in their process dimension which is the rule. Experience comes from the side represented by the horizontal overhang. effects of the firing of the rule as Now the overhang has been split so that Wisdom is the combination of the vertical stave can become a function knowledge and experience. Thus the rule and with its pre (left) and post (right) is a very precise and singular conceptual conditional input and output positions. element for building up networks of We could just as well see this extension

46 Deep Mathematics and Meta-systems Theory by Kent D. Palmer causality. Our extension of the marks the Universal Turing Machine. In other that functionalizes them sends us words the meta-system is the means for directly into the generalization of the switching between Turing machines, i.e. Turing machine and allows us to presentational systems that produce produce images of arbitrarily complex gestalts. The individual Turing machines systems composed by means of the are coded and written to tape, or if the Gurevich Abstract State Machine universal Turing machine is a Method6. multiprocessor then it might allow different Turing machines to run We can however restrict ourselves to the simultaneously as different threads. A production of a traditional Turing multi-processing Universal Turing machine then this would be effected by Machine is equivalent to an operating combining a Turing tape (infinite on one system which is one of the best end and finite on the other) with a ruleset examples of a Meta-system as opposed of functional marks. On the tape would to a System that acts like an application be a series of marks and surreal content. within an Operating System The rules would operate on the marks on environment. In this way it is possible to the tape reading and writing them to see how we can step up quite naturally produce a simulation of a running from the Laws of Form/Pattern level to program. the level of the system without introducing observers from nowhere. State Machine event state Turing state Rule action Also this formulation allows us to Machine consider the computational implications event state state Rule action of the Hyper Kleinian Bottle. In a previous working paper the author has event state state Rule action proposed that we might have Turing Machines with Mobius Tapes. Such a read write TAPE move Turing Machine may simulate an Escher Waterfall and thus simulate a Dissipative System. With a mobius tape the concept

of global non-duality and local duality is Once we have construed a way of added to the concept of local coherence transforming Laws of Form/Pattern verses global incoherence. The tape marks into a dynamic simulation of a takes the place of the trough of the Turing machine then it is possible to soliton. The action of the Turing define very precisely what a System is. machine on the contents of the tape is The system is the showing and hiding of the like the soliton wave. If the wave marks by the operation of the Turing moves around the tape though the action machine program. What ever mark with of the Turing Machine state machine its contents that the Turing machine is then it can come back to the same spot pointing at is the figure on the ground of returning like the water in the waterfall the rest of the tape. A null mark is a over and over again. The difference blank spot on the tape. It is possible to between global incoherence and local go on then to define the meta-system as coherence can be seen in the fact that the marching soliton waves around the 6 http://www.eecs.umich.edu/gasm/

47 Deep Mathematics and Meta-systems Theory by Kent D. Palmer mobius strip will act like solitons or anti- Finally let us go up one more level to the solitons as they pass the same spot on Hyper-Kleinian Bottle. At that level either side of the strip. Duality and Non- there is the shared point of self- duality comes from the fact that the strip intersection between two Kleinian is one-sided and the same in actuality Bottles. This would be represented by even though it appears that it is different two Kleinian Computers that did not locally. know whether they were writing to their own program-data store or the other’s Now let us think about the equivalent of program-data store. We will call this the Kleinian Bottle. We get this figure ambiguous situation Siamese Primal by conjuncting two Mobius Strips. What Computers. This is the level that we lose is the distinction between the Instantatons Super-Breathers arise. We edges and it is inside/outside that can imagine them as the jumping back becomes non-dual. In our picture of the and forth between the two primal solitons moving around the mobius computers that do not know whether Turing machine tape the solitons and they are writing to their own or the anti-solitons would produce static Other’s program-data area. The task breathers out of solitons that just happen switching between these two primal to be opposite each other on the tapes. computers is like the jumping of the At this level we lose the distinction soliton though the potential trough in the between the tape and the state machines field, i.e. the meta-system to another that act on the tape. The tape is outside point in the system. In other words we and the state machine is inside. If we can imagine that the solitons that are lose the distinction between inside and transferred are like agents that move outside, both becoming the same then between different networked data and program becomes the same. We computational environments. These will call the computer that does not agents are not just information but also know the difference between program programs that are transferred from one and data the primal computer after a computational environment to another. construct in Ben Goertzel’s Web Mind7 Siamese Primal Computers are anathema software. to Software Engineering. They are instead the stuff of Artificial Intelligence The Primal Computer is an image of the that has been excluded from Software Autopoietic System defined by the Engineering because self/other rewriting Quaternions, Breather Solitons and code is forbidden in normal software Kleinian Bottles. Such a computer does applications due to its indeterminate not know the difference between the nature. inside (state machine) and outside (tape). It has come to embody global The self/other rewriting software system incoherence even though it may have is a picture of the reflexive social level nodes of coherence locally. Its that are defined by the Quaternions, the inside/outside is non-dual globally while Hyper-Kleinian Bottle and the Super- it is locally dual. Breather. Such a computer is completely mixed up not knowing the difference between self and other within the milieu 7 cf Inteligenesis Company headed by Ben of the meta-system. In fact it is difficult Goertzel See Wild Computing

48 Deep Mathematics and Meta-systems Theory by Kent D. Palmer to tell such a computer from the event and data). These minimal methods computational environment because it is and their application in Software composed of agents that are continually Engineering has been explicated in Wild transferring themselves form point to Software Metasystems by the author. point in the environment spontaneously. The point that we would like to breach It is as if the cells of a cellular automata here is the fact that paradoxes appear in were no longer fixed but could move in the field of minimal methods. These relation to each other freely while at the paradoxes become artificial intelligence same time computing their neighborhood techniques. It was confusing for some to give a value and also a vector of their time why AI did not have methods like own movement. Such a media would Software Engineering had. Then give extremely complex chaotic patterns. eventually I realized that it was because These patterns may be seen to be each AI technique was actually a analogous to the octonion aambrot set paradox in the minimal methods layer that was hypothesized to exist by the and was excluded from Software author and first produced by Onar Aam. Engineering for that reason. Examples of The Mandelbrot set is the most complex such paradoxes are: known mathematical object. It records  Self-rewriting Code the propensities in the field that connects  Expert Systems the real and the imaginary numbers.  Simulated Annealing These propensities appear as "lines of  Neural Nets flight" which appear when a given point  Genetic Algorithms is iterated in this field. Similar structures  Etc. exist for the quaternion and octonion. Each of these is a programming The quaterbrot and the Aambrot (or technique that exhibits some opacity to Octobrot) are multidimensional fractal the human intellect. To our intellect structures. These define the propensities cognition is transparent. But the at these higher levels of complexity workings of these techniques is opaque related to the Autopoietic and Reflexive in some sense. What is interesting is that special systems respectively. we use these techniques as our examples of artificial intelligence. Thus artificial intelligences is naturally opaque to The Artificial human cognition. This fact is extremely interesting because it means that when This brings us to consider the nature of we begin developing artificially Artificial Intelligence, Artificial Life and intelligent agents perhaps by Artificial Sociality in general. We have combinations of these techniques we defined some very strange computational will not understand how they work nor structures which we have called the how they reach the conclusions they primal computer and the Siamese primal reach. This opacity means that these computers. In general these kinds of agents will in fact be Alien to our own structures are outlawed from Software form of cognition. When we imagine Engineering. Software Engineering that we will create artificially intelligent applies the minimal methods that arise agents that will inhabit the realm of between the four viewpoints on every cyberspace then we are really talking real-time system (i.e. agent, function, about production of an alien culture of

49 Deep Mathematics and Meta-systems Theory by Kent D. Palmer agents in cyberspace, the first super Emergent Meta-Systems Modeling multi-media medium, which will interact socially and will have opaque Once we realize that the Turing Machine intelligences and will exhibit artificial is the means of defining the system and life in that they have the illusion of being that the Universal Turing Machine is the alive due to their motion thorough means of defining the meta-system then cyberspace and their transformative we can begin to understand the problem functioning. of modeling the meta-system itself. For that purpose we think Systems Artificial Intelligence is opacity of Dynamics is the best candidate. Systems programming techniques that are Dynamics gives a very accurate picture paradoxes in the software layer defined of the structure of the meta-system with by the minimal methods between the its positive feedback loops that can go four methodological viewpoints. off in either a positive or negative Artificial Life is the independent direction which are balanced against the autonomous functioning of these negative feedback loops that maintain artificially intelligent agents. The fact homeostasis. Homeostasis is the nature that these agents are opaque to our of the autopoietic system. Positive cognition means that they are feedback is the nature of the meta- autopoieticly closed to us as observers. system. Reflexive systems are in a meta- Thus Artificial Intelligence conjuncted stable configuration between negative with Artificial Life automatically yields and positive feedback. Autopoietic Systems. When such systems interact socially then they When we think about the meta-system automatically become reflexive and thus and modeling it; it is necessary to exist at the level of reflexive autopoietic understand that the systems dynamics systems. Thus Reflexive Autopoietic equations are primarily applied to the Systems Theory is the fundamental dynamics of side-effects in the theory of interacting intelligent agents in annihilation or cancellation process. The the meta-system of cyberspace. emergent meta-system is always Cyberspace is a world wide operating canceling. But it does so in an system for artificially intelligent social environment where the associative and living agents. In defining that "region" commutative properties have been lost. the mobius Turing machine, primal Because of that side-effects appear and computer (i.e. two conjuncted mobius those side effects form cascades. When Turing machines) and the Siamese the cascades of annihilations loop then primal computer pairs (i.e. four we have what has been called by the conjuncted mobius Turing machines) are author the anti-category of the archetypal. They occur in the meta- annihilation mosaic. The annihilation system or general economy of the global mosaic answers the question how agent operating system that encompasses everything can be continually the earth through the electronic annihilating yet things seem to persist networks. over time despite radical discontinuity. This is because side-effect cascades loop and these loops produce seeming persistence in the face of continual

50 Deep Mathematics and Meta-systems Theory by Kent D. Palmer relentless annihilation. Notice that this Formalism Specification Appendix continual annihilation is the ever present fact of radical discontinuity which is a In this section we will attempt to give a stronger precondition than entropy definition to our extension to the Laws which is the fundamental assumption by of Form by G. Spencer-Brown. Western Science. Special Systems Theory violates entropy by positing neg- Define Rule called Laws of Form entropy as a possibility. But that violation is based on a model of R1 ()()=() Emergent Meta-systems that can survive R2 (())= radical discontinuity though the looping of annihilation side-effect cascades Define Anti-Rule called Laws of Pattern forming annihilation mosaics. A1 ()()= Emergent meta-systems attempt to A2 (())=() define the count of the swarm members left over after the annihilation process. The elements of the rules are: This count might be seen to be determined using systems dynamics  () - something equations. In such equations the entire  (()) - layering set of difference equations fires like an  ()() - multiplicity expert system in order to determine the count in the next EMS cycle of monads   - null - absence or background within the swarm. If we use Systems Dynamics equations to define these Rules can be applied to expand or counts then we have a way to institute a contract the Form. The minimal model at the meta-system level without expansion is: assuming continuity as normal deployment of systems dynamic Lemma 1 modeling does. In this way we can 1) Start:  model the general economy of the 2) A1:  complete picture which gives our operating system or Universal Turing Lemma 2 Machine a dynamic of its own. The 1) Start:  Meta-system is not passive but has its 2) R2:  landscape of the environment significantly and constantly. The Agents Lemma 1 is equivalent to virtual afloat in this landscape must cope with particles coming into and then popping the continual changes in the availability out of existence. Lemma 2 is a layering of resources governed by the systems popping into and out of existence. dynamics equations that model the articulation of side-effects of We can construct a model of how cancellation. something comes into existence as an alternation of the rules and anti-rules.

51 Deep Mathematics and Meta-systems Theory by Kent D. Palmer

Lemma 3 complex formalism that allows the 1) Start:  arbitrary expansion of these equations 2) R1: () arbitrary collapse back into the 4) >A2: (()) background state of null []. We can 5) >R2:  imagine that we might want to segregate  the application of the rules from the anti- Lemma 4 rules. In that case we would construct 1) Start:  expressions using braces '{}' around the 2) R1: ()() possible to build up something like the 5) >A1:  algebra of the laws of form. However,  our algebra would balance the We notice that Lemma 3 and 4 are immanence of the laws of pattern with inverses of each other. They reverse time the transcendence of laws of form. We and all four rules get used. Between the can already see how these formalisms forward time and backward time can operate independently or in concert. versions something is produced. Prior to There must be meta-rules that define the production of something there is whether we stick to a single Rule set or either layering or multiplicity. After the mix them. Also there must be meta-rules production of something there is either that specify whether we expand or layering or multiplicity. If there is contract the expressions. The multiplicity before then layering comes expressions arise naturally out of the after or vice versa. In other words out of background state and they then again a layering comes something that naturally collapse back into the produces a multiplicity before vanishing. background. If we want to have a series Or out of nothing comes a multiplicity of expansions and then a series of that produces something that produces a contractions then we need to specify in layering before vanishing. We can the meta-rules whether this is allowed or understand the prior and post states to not. The cases we have cited so far move something as the virtual particles prior to toward collapse as soon as possible. It is conservation. We can think of the however possible to construct very layering as the wave state that is the elaborate expressions before we collapse opposite of the particles. If we think of it back again into the background. this way then we see that the (()) is the same as the ()(). It is merely the dual that If we merely apply a single rule set then we see in the wave-particle duality of we hit a minimum expression that does matter that shows up in experiments. not collapse back without retracing its This means that the conserved steps. For instance: something appears by going though a Lemma 5 manifestation of the opposite virtual 1) Start:  properties of matter in our formalism. 2) > expansion ((()))((())) From these humble beginnings it is 4) >A2: ()() possible to see how we could build a 5) >A1: 

52 Deep Mathematics and Meta-systems Theory by Kent D. Palmer

of the combined dual categories we will Lemma 6 continue to elaborate the formalism. We 1) Start:  want to introduce content into the forms. 2) > expansion (()()()()) 4) >R1: (()) Surreal Numbers as the content of the 5) >R2  forms. This is done by using the up and down arrows of surreal numbers. These Notice that we can arbitrarily expand could be seen as substitutes for Yin and steps three in Lemma 5 & 6 but to return Yang signs. By introducing these content to the ground state we must reverse our diacritical marks we are distinguishing steps unless we want to pass over to the quantity from quality. Quantity opposite ruleset. The basic dynamics is participates in a N^2 matrix while that the Anti-rules produce infinitely quality is formed from the deep nesting of two nodes while the interpenetrations of the N elements Rules produce a multiplicity that grows giving a 2^N structure. We notice that in powers of two. the laws of form quantitative structure normally increases at a rate of 2^N. The In Laws of form we encounter arbitrary layering laws of pattern structure series of symbols and have the increases linearly by just adding two possibility of an algebra if we introduce layers each time. We introduce this variables. But it is instructive to see what qualitative structure by adding within the happens merely as we arise from the marks either up or down arrows. ground state itself. Without alternating between rule sets we must retrace our ( \/ ) or ( /\) steps in the collapse from an arbitrary ( \/) \/ ) or ( \/ ( \/ )) expansion. We might collapse back in a Each layer can have its own qualitative different pathway than our expansion marker. These markers may be as and that is when we produce the complex as necessary to express the theorems of the laws of form. Each such quality. Normally though the qualitative theorem has its laws of pattern dual. This layering should match the level of is because laws of pattern is the dual multiplicity. But it may be simplified to category to laws of form. We will not any heuristic level. work out those theorems here as they have already been worked out by G. So for instance ( /\ ) or ( \/ ) is the first Spencer Brown and by Varela and level with a single yin or yang marker Kauffman in their ingenious form This is adequate if there is only one form dynamics formulation of Laws of Form. in multiplicity. What they did not realize was that there was a dual category that reverses the ( \/ \/ ) or ( \/ /\ ) or ( /\ \/ ) or ( /\ /\ ) rules and that the super-category that includes both duals is far more This is the second heuristic level that is interesting in its dynamics than the adequate if there are two forms in the single Laws of Form algebra. multiplicity. We know that these two forms can just pop into existence: () () Instead of covering that old ground again of the development of the algebra If we interpenetrate them then we get 2^N or 2^2 = 4 states. These four states

53 Deep Mathematics and Meta-systems Theory by Kent D. Palmer correspond to the for possible qualities ( \/ /\ ) ( \/ \/ ) => ( /\ /\ ) that are represented by the combinations ( \/ /\ ) ( /\ /\ ) => ( \/ \/ ) ( \/ /\ ) ( /\ \/ ) => ( \/ /\ ) of up and down arrows. ( \/ /\ ) ( \/ /\ ) => ( /\ \/)

So we might show the appearing forms ( /\ \/ ) ( \/ \/ ) => ( \/ \/ ) with their qualities as they appear: ( \/ /\ ( /\ \/ ) ( /\ /\ ) => ( /\ /\ ) ) ( /\ /\ ). ( /\ \/ ) ( /\ \/ ) => ( /\ \/) ( /\ \/ ) ( \/ /\ ) => ( \/ /\ )

There is a transformational rule about This same kind of patterning applies to heuristic qualities which is odd becomes the nesting. Except there we would even and even becomes odd as seen in expect it to reverse. Ilm al Raml (the science of the sands): ( /\ /\ ( /\ /\ ) ) => ( /\ \/ ) o + o = > o ( ( /\ /\ ) /\ /\ ) => ( \/ /\) oo + oo => o o + oo => oo We can express non-associativeness by oo + o => oo allowing the result to be inverted if the or associated layering is on the right as opposed to the left. We can express non- /\ /\ => /\\/ commutativeness if we allow the result \/\/ => \//\ to flip when we cross back though the /\\/ => \/\/ \//\ => /\ /\ arrow.

( /\ \/ ) => ( \/ \/ ( /\ /\ )) Yang Major becomes Minor Yang ( \/ /\)) => ( ( /\ /\ ) \/ \/ ) Yin Major becomes Minor Yin Minor Yang becomes Major Yin These complex qualitative dynamics Minor Yin becomes Major Yang gives a depth to the laws of form previously unavailable. This series of This phenomena is called the rolling heruistics increases by steps as we over of the opposites. interpenetrate more and more monads:

It is possible from the given pair of 1) 2^1 Yin/Yang bigrams ( \/ /\ ) ( /\ /\ ) what the 2) 2^2 Major and Minor Yin\Yang intersection of the qualities would be. 3) 2^3 Trigrams 4) 2^4 Ilm al Raml (Arab divination system) We note that the first monad is odd 5) 2^5 Yin/Yang by 5 hsing (transformations) while the second one is even. Odd and 6) 2^6 I Ching (Chinese divination system) even give us even. Therefore the result is 7) 2^7 Bei (pacific island divination system) the flip of the even or ( \/ \/ ). These heuristic levels move to infinity ( \/ \/ ) ( \/ \/ ) => ( /\ \/) => ( /\ ) down the Pascal Triangle and represent ( \/ \/ ) ( /\ /\ ) => ( \/ /\ ) => ( /\ ) interpenetration of all things ultimately. ( \/ \/ ) ( /\ \/ ) => ( /\ /\ ) => ( \/ ) ( \/ \/ ) ( \/ /\ ) => ( \/ \/ ) => ( \/ ) But we can only handle about this level of complexity. Thus we simplify from ( /\ /\ ) ( \/ \/ ) => ( \/ /\ ) all possible permutations of natural ( /\ /\ ) ( /\ /\ ) => ( /\ \/) opposites to a level that sufficiently ( /\ /\ ) ( /\ \/ ) => ( /\ /\ ) characterizes the level of independent ( /\ /\ ) ( \/ /\ ) => ( \/ \/ ) objects we can hold in our short term

54 Deep Mathematics and Meta-systems Theory by Kent D. Palmer memory or less. These heuristic levels especially the recently discovered ones. are the basis of Chinese and Islamic All of these dynamics may be traditional sciences. They represent the approximated by our form-content levels of complexity of interpenetration formalism that connects Laws of of independent objects. We can only Form/Pattern to Surreal Numbers. consider so much interpentration at once. In fact we can consider about as much as We can think of the surreal landscape as can be found in the reflexive system at being four dimensional. Its four the octonionic level which is at the dimensions are equivalent to the four threshold of seven imaginary things and kinds of patterns: sign, value, process, one real thing. Short term memory is just and structure. We can signify if a big enough to deal with the reflexive particular arrow is from one of these milieu and no bigger. realms by placing a letter in the arrow: \p/, /p\, \s/, /s\, \v/, /v\, \t/, /t\ where 't' Chinese traditional science is based on stands for 'structure'. This could also be cycles taken from the platonic cycles of handled by a different style of arrow for the third and fourth dimensions. We each kind of patterning. In this way we have to treat these platonic solids as can have either pockets of a specific lattices and then we see that they fit kind of pattern separated from another together such that each solid has some pocket: ( \p/ /p\) (\s/ \s/). Or we may level of its lattice that connects to the have mixed pockets of the kinds of lattice of another solid. The lattice of the pattern: (( /s\ /p\ \t/ /v\)). The mixed penthedron of four dimensional space is terms trace a path out into the four crucial because that represents the five dimensional surreal surface space. That hsing and the ten celestial stems. It has surreal space can be seen as the nature of the same group A5 as the icosahedron Wild Being. As we have said previously which has a cycle of 60. The in another context. Surreal numbers are icosahedron has the 12 that relates to the composed of determinate, infinities, twelve branches. All these cycles used infintessimals and holes. If we multiply by Acupuncture that are found in the connect these infintesimals, or infinities, lattices of the Platonic Solids of the third or holes we get a very wild landscape. and fourth dimensions fit into the greater But if we connect either infintesimals or cycle found in J2 the sporadic simple infinities with holes then we get the group that has 604800 members. rudimentary dissipative structures because series of numbers come out of It is interesting that by combining the nowhere. From this we can build up Surreal Numbers and treating them as autopoietic and reflexive structures out diacritical marks that indicate qualitative of these dissipations. Therefore surreal states we can approximate the relation of space which is an image of Wild Being form to content and that gives us some gives us directly a model from which we insight into the formalization of the can derive the special systems. The heuristics of natural opposites. An forms that occur with Laws of Form excellent source for the definition of (rules) or Laws of Pattern (anti-rules) these dynamics is The Meaning of Man merely give us the means to mix this by Sidi Ali al-Jamal [Diwan Press]. Also spacetime partitioning it and one can refer to the Taoist classics manipulating it. This manipulation has a

55 Deep Mathematics and Meta-systems Theory by Kent D. Palmer

Pure Presence aspect given where we and process is the opposite of structure if draw our distinctions in the Wild Being the arrows are marked. So (\s//p\) substrate of surreal spacetime. The cancels (\t//v\). Process Being aspect appears in how far it is between opening and closing The next step is to add on Matrix Logic. parentheses. The Hyper Aspect has to do This is done by giving superscripts to the with the jumping into a certain point in expressions and placing in brackets an expression from another point. It is those that have the same aspect R0,-1 the hyper jumps that cause the mixing. designation: [(\s/)] the possible We can represent this jumping by having superscripts are R for reality, T for truth, 'from' and 'to' hyperlinks in the I for identity, and P for presence. The expression: (((f1)((t2))((()(f2)) t1)). Or if vector positions can take on the values we use marks Spencer-Brown suggests of 1 for asserted, 0 for denied, and -1 for lines underneath the expression that has hidden. Matrix Logic has a complete an arrow. If we are to jump then we must logic for manifestation. It takes the be jumping though one of the other positions of the game of what hand holds dimensions so the hyperlinks justify the the coin. There are the following interpretation of four dimensional surreal positions: space. If the Surreal Space is four Right hand full and out = x,1 dimensional then it is clear that we also Right hand empty and out = x,0 need four dimensional marks. We have Left hand full and out = 1, x Left hand empty and out = 0,x already suggested the notation of the Right hand empty and Left hand full both out = 1,0 rotated marks for this. But if we are Right hand full and Left hand empty both out = 0,1 Both hands full and out = 1,1 expressing the marks in a row using Both hands empty and out = 0.0 parentheses then the notation becomes Both hands hidden = -1,-1 Right hand hidden Left hand Full and out = 1,-1 cumbersome. In stead of this we can use Right hand hidden Left hand Empty and out = 0,-1 four expressions one for each axis in the Left hand hidden Right hand Full and out -1, 1 Left hand hidden Right hand Empty and out -1,0 four dimensional space. The marks represent the segmentation of that axis. Matrix Logic uses truth matrices on aspect vectors to manipulate them. S: (())(()).(())()()(()) for the sign dimension P: ()()(())().(())((())) for the process dimension Vectors may take the Bra or Ket V: ()()()().()()((())) for the value dimension orientation. Matrix operations may T: (())()().((())) for the structure dimension operate on other matrices as well as the vectors. A matrix operation on two A period represents the zero point on vectors may yield either a aspect scalar each axis. Four expressions with periods or a matrix. Reduction to a scalar or would completely segment the four expansion to a matrix are seen as duals dimensional Surreal space. in the meta-system.

We will also allow Forms and Anti- A given expression may have four forms cancel each other. When we have associated superscripts [()()(())] R0,1, T0,-1, a form ((/\/\)\/) then we will assume that P1,1, I0,0. The aspects form another four the anti-form is (/\(\/\/)). When ever dimensional space where the different duals like this occur they cancel each dimensions are manifestation lattices. other leaving the void. For these Matrices may be applied to mixed sets of purposes sign is the opposite of value aspects. This gives the organizational

56 Deep Mathematics and Meta-systems Theory by Kent D. Palmer power of meta-logic to the Autopoietic System. When something is asserted and All this has described a formalism that denied at the same time this is para- allows us to describe the Autopoietic consistency. When something neither System in detail starting from the asserted or denied at the same time this patterning level (structure) and moving is para-completeness. When assertions though autopoietic form to end up at the and denials are both hidden then we system level (organization). It is have the thing existing in potentia. These complex and not worked out in detail. may be true of any of the aspects of But this sketch allows the causal Being. observer to see that it is possible that by interlocking these formalisms we can Since the kinds of pattern arise as pre- construct a model of the autopoietic entities from the aspects of Being we system that goes well beyond what have a closed circle here. Pre-entities Spencer Brown proposed and Varela and take the form of sign, value, process, Kauffman elaborated. structure and these appear in a four dimensional surreal space. This space is Organization describes the manifestation a rhizome of multiple connections. The of the forms with their contents within space is segmented by marks. The marks Being. Being includes the four aspects must be four dimensional to handle the (reality, identity, truth and presence). segmentation of the space. But their four These define the relationship between dimensionality is quaternionic because system and meta-system in the sense that they have to produce autopoietic unity at the meta-system is a filter on the systems the level of form. But these forms that it encompasses. The meta-system is participation in manifestation and continually testing the reality of its manifestation is segmented by the systems while the systems are aspects. Thus we get four different continually attempting to establish their aspect designations that are manipulated truth. The systems are concerned with by the logical matrices. Autopoietic their own identity while the meta-system organization comes from the rings of is concerned with the presencing of the operators Stern describes. Because the system within the meta-system. Thus the aspects of Being are themselves meta-system is concerned primarily with quaternionic that allows there to be physus while the system is concerned autopoietic unity at the system level. primarily with logos. The four aspects of Reality, Identity and Truth are the reality are their means of mutual quaternionic imaginaries within Being. appropriation within the overall structure So the different aspects again gain unity of Being which has four aspects and four within manifestation at the system level, kinds. As we go up the meta-levels of not just the level of the autopoietic form. kinds of Being we encounter at each level a different interpretation of each If we introduce observers to the level of aspect of Being. Reality is different and the autopoietic system it would use has emergent aspects at each meta-level either the Observer Mechanics or of Being. The same is true of all the Jumaries Information, Subject, System aspects of Being. or some mixture of both. Both of these formalisms relativize observers. Jumarie is the preferred formalism for observers.

57 Deep Mathematics and Meta-systems Theory by Kent D. Palmer

As was mentioned in the main body of the paper, in order move to the level of system we need a different model from the generalized model of manifestation that we achieve by employing Matrix Logic with regard to each aspect of Being. Organization is the ordering within manifestation of the forms with their associated contents. But to move to the level of system we need to develop the idea of an object out of the lower level concept of the ‘mark’. We have achieved this by making each mark a rule or function which as part of its processing in time produces a result. Then we can employ the hyper-jumps that G. Spencer-Brown suggests as the means of creating recursion and though that all the other programming constructs. We then concatenate these rule-marks to produce a Turing Machine as previously described. That leads us into the series of computational unfoldings that mirrors the hierarchy of the special systems on up to the meta- system level which is modeled by the universal Turing Machine.

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