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Mathematical Meaning: On the Meta-anomaly
Exploring the Meta-anomaly in Mathematics and its
relation to Special Systems Theory and Schemas Theory
Kent D. Palmer [email protected]
http://kdp.me
714-633-9508 Copyright 2017 KD Palmer
All Rights Reserved. Not for Distribution.
MathMeaning_01_20170315kdp02a corrected 2017.03.15 was MathMeaning_01_20170209kdp01a 2017.02.09-10
Draft Version 02; unedited
http://orcid.org/0000-0002-5298-4422
http://schematheory.net http://emergentdesign.net
Key Words: Mathematics, Systems Science, Meaning, Special Systems, Schemas, Meta-anomaly, Foundations
Abstract: Meaning in Mathematics is interpreted based on the relation between the meta-anomaly in Mathematics and Schemas Theory which includes Special Systems Theory
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discovered via the mathematization of Systems Theory within Systems Science. In particular we want to discuss Special Systems Theory and Schemas Theory as an example of a discovery of some of the meaning within certain parts of Mathematics that I call the Meta-anomaly .
The Meta-anomaly is a series of anomalies in Mathematics and a few in Physics that have a peculiar structure. These anomalies together give us analogies for the structure of the Special Systems and give us further hints about the mathematical basis for Schemas Theory. In order to elucidate this connection I will tell the story of the discovery of the Special Systems and the creation of Special Systems Theory as a counterpart of General Systems Theory within Systems Science. The locus of this work was a project I had to study the structure of the Western worldview called The Fragmentation of Being and the Path beyond the Void1. In that book I explore the
structure of the Western worldview and toward the end of that book I decide to read Platos Laws the first book of Systems Theory and the first Sociology Book, which is hard for a Sociologist like myself to resist. But as I studied this book that is part of our tradition that almost no one reads, I discovered that there were three cities that were all quite odd but which were related to each other which was the Ancient Athens as described by the Republic of Plato, Atlantis, and the city of the Laws: Magnesia2. Being
inspired by Systems Theory I treated these peculiar imaginary cities in a systematic way together and noticed that they had some strange properties. And then at one point I thought, I should look for these same structures in Mathematics. And immediately I found a match to that structure in Hyper Complex Algebras. Then as I searched further I found further examples of analogies in Aliquot Numbers, Non-orientable surfaces, but also in some analogies in physics like solitons, super-conductivity s Cooper pairs, and Bose-Einstein Condensates. Eventually I collected several examples of analogies in mathematics for the kind of structure seen in the Imaginary cities of Plato, which was quite odd. So, I hypothesized the Special Systems as a branch of Holonomics and specified three special systems called: Dissipative Ordering, Autopoietic Symbiotic, and Reflexive Social3. Each holon4 was associated
with a theory that most closely approximated the nature of the various special systems. Dissipative Ordering Structures were associated with Prigogine5 and his
negative entropy6. Autopoietic Symbiotic Special Systems were associated with the
theory of Maturana and Varella7. Reflexive Social Special System was associated with
1https://works.bepress.com/kent_palmer/2/ 2https://www.wikiwand.com/en/Laws_(dialogue) 3https://works.bepress.com/kent_palmer/3/
4Koestler, Arthur. Janus: A Summing Up. Nueva York: A Division of Random House,
1979. https://www.wikiwand.com/en/Holon_(philosophy)
5Prigogine, Ilya, and Isabelle Stengers. Order Out of Chaos: Man's New Dialogue with Nature. London: Fontana,
1988.
6https://www.wikiwand.com/en/Negentropy
7Maturana, Humberto R, and Francisco J. Varela. The Tree of Knowledge: The Biological Roots of Human
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the theories of the Reflexive Sociologists like John O Malley8, Barry Sandywell9, and
Alan Blum10. Once I had the overall meta-theory taken from Mathematics based on
mathematical anomalies and candidate representative theories taken from Physical, Biological and Social Sciences then Special Systems Theory was born. But what I found most fascinating was the fact that each of the different mathematical analogies contributed a different understanding of the Special Systems themselves. The anomalies seemed to work together to define the special systems very precisely, more precisely than any one anomaly in mathematics did on its own. Thus was born the idea of the Meta-anomaly in which there are structures that are mirrored around mathematics and dispersed in different types of mathematics or physical phenomena that were very much a like such that taken together they indicated the possible existence of Special Systems that were anomalous in relation to General Systems Theory such as that provided by Klir in Architecture of Systems Problem Solving11,
which is one of the most mathematical of the different General Systems Theories. There is this uncanny resemblance between the various anomalies that make up the meta-anomaly despite the extreme difference between the forms of mathematics involved, the overall structure is very similar, but the differences tells us new information about the Special Systems that we would not know otherwise, than through the insights give us by finding very different mathematical analogies with a global similarity to each other. What followed was years of research attempting to find precursors in various cultures for these Special Systems. And precursors were found scattered about in various very different cultures. And slowly Special Systems Theory grew and the number of precursors grew and the mathematical analogies grew until I thought I had a complete theory. This theory is described in other papers12. Here we will only mention parts of the theory that is important to our
argument in this paper. Eventually in order to try to understand better the nature of Special Systems I developed Schemas Theory (http://schematheory.net), and after that a worldview theory13 in order to give context to Special Systems Theory and to
understand better what it meant. Always the watch word was to follow the mathematics where it led with as few preconceived ideas as possible as to what Special Systems were. Eventually this became a very robust theory which was summarized under the title Dagger Theory14 which includes Philosophical Principles
of Peirce and Fuller, Foundational Mathematical Categories, Ontological Schemas, and Epistemic View-Order Hierarchies. The aim was to give a ground to Systems
8O'Malley, John B. Sociology of Meaning. London: Human Context Books, 1973.
9Sandywell, Barry. Reflexivity and the Crisis of Western Reason: Logological Investigations. Place of publication
not identified: Routledge, 2013.
10Blum, Alan. Theorizing. London: Heinemann, 1974.
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Architectural Design15 and as a side effect to Systems Engineering16. And I believe that
this research program is finally achieving many of its goals.
But what I want to talk about here is the way in which Special Systems Theory and Schemas Theory gives meaning to Mathematics which is unlike any other theory that I know about, and which I think is an unexpected result, i.e. that certain types of mathematics have a nondual meaning (http://nondual.net). Mathematics is usually thought to transcend our uses of it and of itself to lack a specified meaning. It is pure order which has significance within its own terms but no meaning that goes beyond its purview. But the approach of Special Systems Theory is different from the way that most theories are built. )t started with an example from the tradition, Plato s Imaginary Cities that were systematically being compared and contrasted. But then I looked for mathematics which were analogous to the structure of these Imaginary cities taken together, and found that all that were isomorphic to the Cities of Plato were Mathematical Anomalies. When I saw that the various Mathematical Analogies brought to the understanding of the cities I found that each one brought a different type of intelligibility to it. And also that the various different mathematical analogies interlocked to give a more precise meaning to the various special systems separately and taken together as well. And being a Systems Theorist I generalized to produce an image of abstract systems that had the mathematical properties suggested by the Mathematical analogies. I considered an example or prototype to be any system that had the properties of the various kinds of Mathematical Analogies that were isomorphic to the Cities. In other words, I generalized from Cities to Systems with those mathematical properties of which Plato s cities were merely one example. Then I went searching for other examples and found them. For example, Herodotus has similar types of structures in his description of Babylon, which we know are not a description of the actual Babylon City17. For instance, Homeopathy18 is an example of
a kind of traditional medicine which is analogous to the Dissipative Ordering Special System, and Acupuncture is a kind of traditional medicine which is analogous to the Autopoietic Symbiotic Special System. Thus, in China I found a very well fleshed out theory of how Autopoietic Special Systems work, and it is associated with a kind of medicine that is considered efficacious, unlike Homeopathy19. But also, I discovered
that if you combine various Special Systems with a Normal System then you can get a structure I call the Emergent Meta-system. And I discovered that a model for this is the game of Go (Wei Chi) in China. And what was strange was that this model via the game of the Emergent Meta-system is more precise than the mathematical models in
15
https://www.academia.edu/31797031/Software_Systems_Architectural_Design_Foundations_01_Introduction
16https://www.academia.edu/31038671/Foundations_of_Systems_Architecture_Design
17Kurke, Geoffrey. Coins, Bodies, Games, and Gold: The Politics of Meaning in Archaic Greece. Chichester:
Princeton University Press, 1999.
18Hahnemann, Samuel, and Constantine Hering. Organon of Homoeopathic Medicine. Charleston, S.C:
BiblioLife, 2010.
19Ameke, Wilhelm, and R E. Dudgeon. History of Homeopathy: Its Origin, Its Conflicts. New Delhi: B. Jain Pub,
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many ways. So I looked far and wide for examples of the Special Systems and found them scattered over the globe here and there in various cultural artifacts from very different societies throughout history. The search for precursors had the goal to make sure that these kinds of systems actually existed and were recognized by others in various societies. But the greater goal was to apply Special Systems Theory to something closer at hand which was understanding the nature of Architectural Design of Systems, like Software Systems or at higher levels of abstraction in Systems Engineering. I wrote a dissertation on this in Systems Engineering called Emergent Design (http://emergentdesign.net). Eventually I developed a tutorial about Schemas Theory which I delivered before the INCOSE.org and ISSS.org conferences in 2014 (http://schematheory.net). Various papers on Schemas Theory, Meta-systems Theory, and Special Systems Theory have been given at CSER, INCOSE, and ISSS conferences (http://archonic.net). Up until the present this research has continued with a recent presentation on Schemas Theory to the INCOSE.org Systems Science Working Group (Jan. 2017)20.
What we would like to concentrate on here in this article is the way that a global structure in mathematics which I call the Meta-anomaly gets meaning by drawing anomalies from various parts of mathematics that are similar and by creating a Special Systems Theory that is wholly based on the mathematical properties of those so called Platonic objects. Special Systems contributes by drawing together various anomalous mathematical structures that otherwise would not be seen as related. Special Systems benefits by the fact that the various kinds of mathematical entities seem to lock together to describe the Special Systems in a great deal of detail not available from one type of mathematics alone. Special Systems had only one example that served as prototype in the beginning which was Plato s )maginary Cities. But then once the abstraction was formed of the Special Systems as a generalization of anomalies of a certain type then other exemplars were found that had a similar structure of various types. The confidence in the existence of Special Systems was improved by the finding of precursors in various traditions. But the theory itself was used to attempt to answer the question of the nature of Consciousness, Life and Sociality. It was found that the theory had its closest analogy in the theory of Anti-Oedipus21 and Thousand Plateaus22 of Deleuze and Guattari. Also the work of
Terrence Deacon called Incomplete Nature23 is very close to the spirit of the Special
Systems approach, but Deacon has no mathematical basis for his theory. Special Systems Theory is totally driven by its mathematical analogies based on mathematical anomalies taken from the Meta-anomaly that produces fusions in peculiar ways in mathematics as numbers get closer and closer to one. We see the Meta-anomaly as a fusion as numbers approach one from infinity that creates various anomalies that reconcile finitude with infinity. But in this process we find that the
20
https://www.academia.edu/31086576/Systems_Philosophy_Questions_concerning_Schemas_Theory_Answered
21Anti-oedipus : Capitalism and Schizophrenia. University of Minnesota Press, 1998.
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Meta-anomaly itself has meaning and that meaning is rooted in nonduality. There is a model of nonduality implied within the meta-anomaly that is unexpected (http://nondual.net). This suggests that there is meaning in the meta-anomaly itself that is not projected there by the interpreter because of the oddity and the precision of the mathematical structures that are nondual in their import. And this is what I want to suggest is the most interesting thing learned from the exploration of the possibility of Special Systems Theory and its mathematical analogies.
This is a completely different way to read the mathematics. We look for anomalies in mathematics that have a similar structure. Then we notice that they together define something together that cannot be defined as well by them separately. Then we create the image of Special Systems as a general systems theory of these kinds of anomalous systems. Then we look for examples of those systems other than our original prototype (the imaginary cities of Plato). For instance, we find other examples of them in Plato, for instance in the Symposium, or in the description of Babylon by Herodotus. Then we find examples of them as well worked out practices as in Homeopathy, or as theories as in Acupuncture in other cultures or in the Western tradition, as in Alchemy. Then we use the exemplars to augment our understanding of the theory which as it becomes more precise allows us to find other examples, until we find examples like the game of Go (Wei Chi) that is even better than the mathematical models. Or, for instance, the only book that seems to be a self-conscious description of the Emergent Meta-system is the commentary on the Awakening of Faith by Fa Tsang24. Or we can see the mudras25 of the five Buddhas as examples of the various
Systems and Meta-systems along with the Special Systems. These cultural products that mirror the Special Systems are found throughout history in various cultures. Since the Special System have an unusual structure it is fairly easy to see whether seeming isomorphisms are really examples of them or not. And thus slowly our knowledge grows as we find more precursors or understand the mathematics better or find new mathematical analogies. And of course, in the process we set up hypotheses based on the mathematics and see whether these hypotheses hold as we find new examples. Slowly we build up a science of Special Systems based on the hints we find in Plato together with structures of order we find in various disparate places in mathematics that seem utterly unrelated to each other. But slowly a picture of the Meta-anomaly arises which posits that these various mathematical anomalies were actually always related to each other and portray different aspects of the same thing in various different forms of order with different types of mathematical elements. And of course, we find other confirmations such as the idea of Concrete Universals26 in
Plato and in Hegel in which they seem to be theorizing directly about the Special Systems. For instance, there is internal evidence that Leibniz knew about the
24Fa, zang, and Dirck Vorenkamp. An English Translation of Fa-Tsang's "commentary on the Awakening of
Faith". Lewiston N.Y.: E. Mellen, 2004
25Saunders, E D. Mudrā: A Study of the Symbolic Gestures in Japanese Buddhist Sculpture. Princeton, N.J:
Princeton University Press, 1985.
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Emergent Meta-system and was describing it in his Monadology27. But these
examples of conscious use and appearance in philosophy are rare. By learning about the mathematics and looking for precursors, prototypes or exemplars we refine our concept of each of the special systems and their relations to each other. And slowly this turns into a way of seeing things in general and eventually that changes our way of looking at the world. We begin to see the world as embodying special systems. By learning a mathematical language about anomalies we eventually begin to see them in the world and see the world itself in different ways than we would have before finding out about the Special Systems, and then that means that we recognize them faster, and with each one we learn more about the variety of embodiments the special systems can take, many of which are unexpected.
I want to suggest that this way of using Mathematics ascribes Meaning directly to it eventually rather than merely projecting interpretations on it due to its alignment with phenomena. What we begin to see is that certain anomalies in mathematics are themselves models of nonduality, and they were meant to be that from the beginning because they embody the structure of nonduality itself directly. And this gives us a different way to think about the Platonic realm and whether mathematics is a convention invented and constructed by humans or something there from the beginning to be discovered. To the extent that we can construct notation and mathematical categories however we deem best makes them seem constructed, and to the extent we cannot change the order relations of these objects regardless of notation makes them seem fixed externally to ourselves and therefore discovered. There is a sense in which any intelligent creatures would have to discover the same mathematics even if everything was called something different and appeared in different orders of discovery historically.
An understanding Schemas theory helps in this regard because the Schemas give us something to count. So, in some sense Schemas come before mathematics. Schemas are a priori projections of spacetime envelopes as templates of intelligibility for experience. It is Schemas that we count at various scopes but within the Schemas there is an indication of the nature of Special Systems that has its mathematical analogies which define them working together. It is the idea that the kinds of mathematics with similar structure from disparate parts of mathematics would work together to give a model of Special Systems that is important. Because of the precision of this coming together of various anomalies in mathematics as if there were a Meta-anomaly that begins to suggest that the mathematics itself has this nondual meaning, that it is not something projected on the mathematics. And this suggests that at least some mathematics has an intrinsic meaning within itself of nonduality that is embodied in the various anomalies we see in the Meta-anomaly. If this is true then we need to reconsider the question of the meaning of mathematics. We have not been looking at mathematics in the right way. It actually does have meaning of its own beyond the intrinsic meaning of its order. And that meaning indicates the nature of nonduality through the various specific anomalies of the Meta-anomaly. And we can
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collect these various analogies together to describe very precisely the Special Systems, and then we can use the Special Systems to find examples in the world of these various types of strange systems that are images of (olons in Koestler s sense. And thus, we build up a science of Holonomics28. Also, we find that Systems have a
dual called Meta-systems and the Special Systems appear in the barzak (in Arabic) or interspace between the Systems and Meta-systems in the fourth dimension. We end up generalizing B. Fuller s Synergetics29 to the fourth dimension30 and then we gain
further geometrical models of nondualty through fourth dimensional analogies. And so it goes in a virtuous hermeneutic circle in which the more we know about the math the more examples we find in various cultures, and the more precursors we find the better we understand the prototypes and exemplars, and also the better we understand the mathematics. And the better we understand the Special Systems the better we understand the limits of General Systems Theory in Systems Science. And the better we understand anomalous phenomena like Consciousness, Life and the Social. Special Systems becomes a kind of Rosetta stone for unraveling the nature of the worldview and understanding the relation of various worldviews to each other through the precursors of the Special Systems that are found within them.
This gives a completely different way of thinking about Platonism in Mathematics. What we see in Plato is Egyptian Wisdom repackaged for Greek consumption. And we can think this because in Egyptian Myth we find examples of structures that are similar to the Special Systems. We find them also in China at a very early date in Taoism, in Acupuncture, and in the game of Wei Chi (Go). And we find them in Buddhism. We also find them in various other places in the Western tradition but they are very rare in our tradition, appearing mostly in Plato and Herodotus as well as Egyptian myth and Alchemy. We can speculate that Alchemy was originally a science of Holonomics in Egypt that was corrupted, but with glimmers of the original theory scattered about for archeological excavation in the history of ideas. Plato was giving us news as Herodotus did before him of this Egyptian science of Holons that they coded into their works as an exoteric doctrine hidden in plain sight in their works. We know that Herodotus went to Egypt. And the fact that the same Special Systems appear in the Histories as the Works of Plato is a major confirmation of the theory. It was not just Plato who had this idea and seemed to get it out of thin air. He in fact tells us where he got it, which was Egypt. And the fact that it appears in coded form in Egyptian myths is a further confirmation that the Egyptians probably had a very well developed science of Holonomics which was eventually turned into Alchemy. Neo-Platonism also preserved this tradition and passed it on to the Italians during the Renaissance. But we cannot really gage what is part of this peculiar esoteric tradition of Holonomics and what is just made up only by finding the mathematics and
28
https://www.academia.edu/3795408/Meta- systems_Engineering_A_new_approach_to_systems_Engineering_based_on_Emergent_Meta-systems_and_Holonomic_Special_Systems_Theory
29Fuller, R.B, and E.J Applewhite. Synergetics: Explorations in the Geometry of Thinking. New York: Macmillan,
1975.
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comparing what we find in the various traditions with the mathematics. Whatever corresponds to the mathematics is designated as real, and whatever does not correspond to this strange rare structure of the mathematics is treated as mere fantasy on the part of various authors. The mathematics works as our filter for seeing the Special Systems within the overall tradition. Once we have the mathematical and physical analogies then we can begin constructing for ourselves models of the Special Systems and we can try to use them to attempt to understand anomalous phenomena like consciousness, life and the social. And then we can go ahead and formulate things like Dagger Theory that contains Philosophical Principles, Foundational Mathematical Categories, Ontological Schemas and View-Order Epistemic Hierarchies as a basis for a deeper understanding of Design Theory and especially Architectural Design in Systems Engineering and Software Engineering.
Suddenly we see that Plato did not have an odd theory of a Platonic Realm as much as a theory of Special Systems that he was trying to explain. And he found many subtle ways of explaining it such as his ideas about Concrete Universals that was taken up again by Hegel in our tradition. We can triangulate what that theory might have been through our study of the mathematical analogies, and then we can attempt to reconstruct that Plato was trying to say to us about the Special Systems in various places in his dialogues. By bringing together later versions of this Special Systems theory from other cultures we can clarify what Plato probably meant. It helps to sort out when Plato is being ironic from when he is describing straightforwardly the Special Systems directly. Because the Special Systems have some odd properties it is easy to misunderstand Plato when he is describing the Special Systems. A lot of cross checking is necessary to sort out the confusions particularly those introduced by the translators because they did not understand what he was referring to. But the interesting idea is that this was coded into the Mathematics and that the Egyptians probably discovered it by exploring the mathematics itself and then through observation of the world by looking for the anomalies in it. Anomalies exist that are similar both in mathematics and physics, and thus we can see clearly that these aspects of the Meta-anomaly have real consequences within the world, as anomalous physical structures with anomalous properties. But the essence of these properties appears in four-dimensional space and its geometrical description and is associated with nonduality.
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directly relevant to our study and we can see that mirrored in the Aliquot numbers. Basically we look for structures in mathematics that are analogies for the cities of Plato, and then once we find them we put the various analogies together to construct Special Systems Theory, then we apply that back to find precursors, exemplars and prototypes that then takes us back to looking at the mathematics more closely and attempting to find other kinds of mathematics that are also analogies. Then we find theories and philosophies that have the same structure within our tradition and others and use that to speculate on extensions to Special Systems Theory. But eventually we get back to the fact that the Meta-anomaly seems to have these structures of this form built into it, and thus that these are the intrinsic meaning of these structures that exemplify nonduality and holonomics. And this is where we begin to think that the Meta-anomaly has its own meaning that is this particular meaning and we have misunderstood Plato and what he has been suggesting all these years. Plato is looking at anomalies, and he concentrates on mathematics that mirrors these anomalies in existence and explains them through its ordering structure. When two different types of math are conjuncted they form an autopoietic symbiotic relation with each other that allows us to go beyond the information given in the various mathematical analogies by themselves. And the various types of mathematical analogies are part of a reflexive field of these anomalous structures that can be variously conjuncted to give us different types of information about the Special Systems and Holonomic that flows from their definition. Mathematics itself starts to look differently to us. It appears as if the non-anomalous parts of Mathematics are there to define the anomalies. And the anomalies together define the special systems. And then to understand the Special Systems it is necessary to have Schemas Theory, but that turns out to provide what is countable and it extends our reach to other domains through the templates of a priori intelligibility of spacetime that it provides. And thus we enter a virtuous circle in which whatever we find out about the math feeds into our recognizing precursors, which in turn leads us back to the math, which then leads us to recognize other aspects of our tradition that we did not appreciate before using mathematics as the key discriminator and following it where it leads. An excellent example of this is August Stern s Matrix Logic. He follows where the math and logic leads and does not project constraints on it but rather learns from it new things. We have been doing similar work as that with Special Systems Theory over the years.
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the non-orientable surfaces that are anomalies within mathematics itself. But catastrophic collapse from plurality producing strange fusions does not square with the idea of creation. But on the other hand, it is very much like the collapse of polytheism into monotheism. In other words, it is much like the actual production of the idea of a single god out of the myriad gods that existed in ancient times in the Mythopoietic era in the transition in theology to the metaphysical era. Let s call this alternative theory that is not thought in our tradition de-creation. Let us contrast this to the ideas of Badiou31 which is that the ultra-one arises from the Multiple, pure
heterogeneity and incommensurability as an event that produces the particulars that are the members of the Set. Badiou is hard pressed to explain how the ultra-one arises as a singularity out of the Multiple. The Multiple is his radicalization of the Assemblage theory of Deleuze. Sets can operate without any particulars within them by treating the null set as zero and the empty set as one. But for something to exist, i.e. to go from ontology to ontic some One must arise as an event out of the multiple. But de-creation is the exact opposite which is a collapse from plurality into one. What if we considered the possibility of an ultra-plurality instead arising from the Multiple. This would be more in keeping with Deleuze s nomadic distribution idea coming before agrarian hierarchies. Out of the Multiple the ultra-plurality as an assemblage arises and then it collapses into One like when we went from polytheism to monotheism. And then the One collapses again into singularity when it is confronted by negation to produce negative one, which is the singularity that is the source of the imaginary numbers. The imaginary numbers are all separate from each other because continuity is negated and discontinuity reigns, and we can see how this discontinuity produces plurality. So here we have a cycle from singularity, to plurality, and plurality collapsing into one, and then from one as a number to a further collapse into singularity by negation, and that negative one then through the square root of negative one produces discontinuity in the imaginary numbers. But also, it produces an infinite series of Hypercomplex algebras, and in the process we get the first few algebras with special and rare properties that are the inverse fusion that lead to the best model of the Special Systems. But when we have one then it is possible to build Pascal s triangle which is the positive image of synthetic unity. But the Pascal Triangle going to infinity gives us plurality again. In this what appears is that there is a cycle that is like the Emergent Meta-system and this appears on the background of the Multiple which is a Meta-system to this systemic cycle. The System is unified and totalized while the Meta-system is disunified and detotalized. But the Multiple is a de-emergent meta-system which is wholly heterogeneous and incommensurable. On this background synthesis appears through the Groupoid structure that can take individual elements and produce syntheses like the simplicies that come out of the Pascal Triangle. Notice here in this de-creational cycle everything happens in a way that is unexpected from the point of view of theory. We are pretty sure that the idea that there is internal creation Spinoza s expressionism and wholly immanent which is opposite the idea of external creation in which God creates something completely other than Himself are nihilistic opposites. Emanationism is a compromise in which
31Badiou, Alain, and Oliver Feltham. Being and Event. London: Bloomsbury Academic, an imprint of Bloomsbury,
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God instead emanates from himself outward to produce creation, as degraded versions of himself. The other possibility which is what actually happened in theological history is that God was created by a collapse of polytheism in the Mythopoietic era into Monotheism in the Metaphysical Era producing Ontotheology and the idea of God as the Supreme Being. So we go from three possible theories in which God is creating everything to a theory about the creation of God. Badiou s idea of the Ultra One arising as a singularity out of the Multiple is just as fantastic as God s creation of everything. Miracle occurs here. The arising of one God out of polytheism is something we know happened. And it collapsed around an anomaly among polytheisms which is the Jewish idea of God as a divinity that you could have a contract with as a people. And as a people the Jews were the only polytheistic religion that affirmed monotheism of their own god as the only real god. And the Jews had a book that they forged in exile that explained how they had broken their contract with their God and that was the reason that history had treated them so badly32.
Concression of polytheism around an anomaly and the production of polytheism in the unlikely form of Christianity is an example of the role of the anomaly in the collapse from multiple to one. Strange things occur such as those seen by Kierkegaard who admits that Christianity is paradoxical and absurd with its idea of incarnation, i.e. the production of avatars like Vishnu s appearance as Krishna. Once we move from the idea of God creating everything to a theory of the creation of God as one god out of polytheism (worship no other god before me, admitting that there is the background of other gods) and we see this process of the catastrophic collapse from plurality into the individual unit, then the production of the meta-anomaly becomes understandable. As we go from many to one there needs to occur internally to the field of numbers accommodations in the fusion process that produces anomalous structures around the area of seven, six, and five, four and three but extends to the whole of numbers in its ramifications as they collapse catastrophically from the cardinal Alephs down to One, just as the polytheistic gods collapsed from many down to one monotheistic God with its own anomalies that occurred in the process. Polytheism collapsed around the anomaly of Judaism into Christianity in the West ultimately producing radical monotheism in Islam. We take the question of how God produced the world, and reverse it into how was God produced within the world, and it is clear that the concept of One God came from the background primal situation of many Gods. Similarly, we can think of the production of one out of many numbers, a plurality, a multiplicity as producing a meta-Anomaly in which strange properties appear from the fusion of the many into one in this symmetry creation as the asymmetrical collapses into symmetry. Various fusions and peculiarities are produced in that process of concrescence. But once we have One then we can negate it, and get negative one. And it just turns out that if we take the square root of negative one we get the imaginary numbers, which takes us from the continuity of the reals to the discontinuity of the imaginaries which is different from the discontinuities between natural numbers. With imaginary numbers there is an infinite progression via the Caley-Dickson process. And in that unfolding of the imaginaries there are special properties in the first few algebras that are produced then are quickly lost.
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But then this progressive bisection of the imaginaries can be emulated using natural numbers and this gives us Pascal Triangle which then is the information infrastructure, but also the generator of simplicies in Geometry through the use of groupoids that produce syntheses. But this is also an infinite progression that takes us back to a constructed plurality that is synthetic. Syntheses are totalities. Unity and totality are here seen as side-effects not as the primary drivers of the process. The primary driver is the collapse of plurality into the individuality of the unit and the aberrations that catastrophic collapse of asymmetry into symmetry causes. Normally we think about the progression from symmetry to asymmetry and the collapse of symmetry into asymmetries, not about the forging of symmetry from Asymmetry, but it is precisely this forging of symmetry that can be seen as responsible for the production of the Meta-anomaly, i.e. a field of very diverse anomalies of approximately the same structure near one, as we descend from infinities down to the individual unit. But once we have the individual unit as the One then it is possible to create other numbers progressively through the negation of operations which ultimately lead to the taking the square root of negative one that produces a discontinuous number field of the imaginaries. And these imaginaries have their special properties that are the best basis found so far for the Special Systems. But that production of imaginaries by progressive bisection goes on forever. But only the first few algebras have interesting properties that are anomalous that gives the basis for the Special Systems. And then when we have the idea of progressive bisection then we can apply it to the natural numbers defined by Peano s Arithmetic to produce Pascal s Triangle that gives us the information infrastructure and the syntheses of the geometrical simplicies and thus dimensionality through geometry. And it is through the arising of syntheses that we actually get the idea of unity and totality. Each simplex at each dimension in the Pascal Triangle in its lattice we can see it arise from unity, and then differentiate, then return to unity. And all the moments of this journey are seen as a totality, as is the synthesis that is produced in the process which is the simplex in each dimension and the production of these syntheses goes on forever through the application of co-recursion. But ultimately this results in the articulation of plurality. And from Cantors paradise which is ultimately produced by the co-recursion there is the possibility of collapse and concretion into an individual unit again which is what gives us the meta-anomaly, which is a field of anomalies with similar structure distributed in diverse places in mathematics. And from studying the field of these anomalies we call the Meta-anomaly then a general theory of anomalies appears we call Special Systems Theory. Then when we look for physical anomalies with a similar structure we find them as well, so the anomalies of mathematics also appear in physical phenomena and thus Special Systems Theory bridges between mathematical anomalies and physical anomalies like Life, Consciousness and the Social.
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which are themselves fused together in our experience and characterize ourselves as living social conscious creatures that exist. The necessary condition for the possibility of our existence appears because there are anomalies in physics and mathematics with the same structure globally though dispersed, they are sparse and rare scattered throughout the sciences as unexplained anomalies. But when we generalize these anomalies and make Special Systems Theory from them then we can use this part of Systems Science as an explanation for the appearance of other anomalies like Life, Consciousness and Social not in their specifics, because each anomalies are different in its specifics, but in general because all these anomalies have roughly the same structure that we can understand through their comparison. Thus, it is a general theory of anomalies that operates on their similarity and resemblance across mathematics and physics and then applies that to solve concrete problems like the anomalous nature of Life, Consciousness and the Social. This is different from the normal General Systems Theories that are based on common or non-anomalous structures that appear in various sciences for instance tat of Klir in ASPS. Instead of asking what all or most systems have in common we ask what anomalous systems have in common, i.e. what is common to the uncommon. And this gives us a different approach to mathematics and physics were we are looking for anomalies and studying their structures and attempting to generalize from what they have in common but also taking their peculiarities and uniqueness into account as possible drivers of properties of Special Systems that are available as bases of explanation across anomalies of different kinds. We are using the basic structure that appears in most anomalies as a way of explaining specific structures in particular anomalies like Life, Consciousness and the Social.
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hyper-efficient and hyper-effective. And these two properties suggest that the field of anomalies has some underlying order to it. And they also suggest that it is the production of one from many that is the source of the Meta-anomaly. And this suggests the hypothesis that these particular kinds of Mathematics have intrinsic meaning, not just projected or interpreted meaning. It is this idea of intrinsic meaning that is anomalous in Mathematics as a whole. And the intrinsic meaning has to do with the modeling of nonduality in mathematics. The anomalies describe holons that are both part and whole at the same time. And these holons have nondual properties, i.e. properties that are something other than merely one or many, but through fusion an alternative is produced in the Meta-anomaly that goes beyond cardinality and has to do with Zero, as emptiness or void. )n other words what is left out of Kant s categorical scheme is Zero. We have as judgements Unity, Totality and Singularity, and this gets mirrored in the Categories as Unity, Totality and Plurality. This assumes that the Singularity in judgement is repeated to produce the categorical plurality. Unity and Totality stay the same. But this does not take into account the fact that these may have opposites, i.e. the disunified and the detotalized, or the indeterminate which is not singular nor plural, i.e. like zero, non-cardinal. What is unified and totalized is the System. What is detotalized and disunified is the Meta-system. A system is a whole greater than the sum of its parts. A Meta-system is a whole less than the sum of its parts, i.e. a whole full of holes like a sponge. Something like the sponge in number theory are Surreal Numbers. The system of numbers (natural, integer, rational, irrational, transcendental) is contrast with the surreal numbers that is a meta-systemic image of numbers that has holes in it. In fact, you can do mathematics with the holes in Surreal numbers. But Number Theory and the Surreal numbers are not commensurable because you cannot do integration with Surreal numbers due to their swiss cheese like constitution being full of holes. But these extremes of the System and the Meta-system revolve around emptiness or void, i.e. the nondual neither one nor many but something else. And this is missing from the judgements and categories of Kant.
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So there is another Notion which Hegel did not know about which is the Notion of the Mass, Instance and Individual. What is shared between them is the individual. And we must attempt to say how the individual arises. One way for it to arise that is not miraculous is by the collapse and fusion of many into one individual. And that produces the Meta-anomaly across the field of math that appears as embodied in many anomalies. Sets have Syllogistic Logic and Masses have Pervasion logic. Pervasion logic was developed in China and India and became the default logic of Buddhism. For the Buddhists Emptiness is a mass concept not a set concept. It is not the empty or null set which as set theory shows are duals of each other. Instead Emptiness as well as Void in Taoism are nondual, i.e. non-cardinal ideas that are neither one, nor two, nor many but something like Zero. So the whole notion has within it two duals Set and Mass that revolve around the individual unit expressed as either particular or instance, encompassed either by Set or Mass and compared to Universal or Boundary. (egel s dialectic always goes from Universal to )ndividual to Particular. But we can conceive of an anti-logic that is pervasive that goes from Boundary to Individual to Instance. Notice that going from Universal to Individual or Boundary to Individual is a collapse, and concrescence out of which is produced the particularity or the instance in each case. But the individual is specified by the fact that it is not empty, nor void but something on the background of nothing. Hegel takes the Judgements and Categories of Kant concerning quantity and generalizes them into the Notion. But he does not generalize far enough to get the co-Notion of Mass. Mass concepts are suppressed in our tradition.
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produces at each binary level a real and an odd number of imaginaries. If we take this progressive bisection and apply it back to the natural numbers defined by Peano s Arithmetic then we get the Pascal Triangle. And that produces instead of discontinuity a series of continuous syntheses through co-recursion and the application of groupoid operations that define an infinite number of dimensions by the simplicies that exist within them. It is in Pascal s Triangle that Unity, Totality, Singularity, Emptiness, and Plurality all are mutually defined diacritically in relation to each other. If we treat the ones at each level of the Pascal Triangle as a set we get the information infrastructure. If we treat them as a mass by assigning undefinables to them like point, line, surface, solid, hunk, etc. then we get the Simplexes for each dimension as a synthesis. Unity appears as the one at the beginning of each simpletic lattice, and totality appears as the one at the other end of the lattice. For instance as seen in the lattices for the triangle and tetrahedron as 1-3-3-1 or 1-4-6-4-1. These simplicies are all self-dual. Plurality appears as the number of individual ones at each place in the lattice. Singularity appears as the ability to isolate each of these units. Peirce calls these firsts. When we group the numbers in each place in the lattice then we get relations which are Seconds that are relata. By assigning undefinable geometrical elements to each position in the lattice we produce continuities that are then used to produce the simplicies as synthetic geometrical figures. When we produce the Platonic Solids of each dimension we can study their synergy and integrity as B. Fuller did for the third dimension. But we can see that the Pascal triangle also has emptiness as the spaces between the numbers that are grouped into the lattice at each level. And we can think of void as what lies outside the lattice prior to the one that is the origin of the lattice. What we see is that the Meta-anomaly is produced by the collapse of the many into one. But the Pascal Triangle is the embodiment of emanationism going from one to the many.
1 11 121 1331 14641 Etc.
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are different from the Real Algebra that can be seen as existing in relation to the Pascal Triangle that stands for the operations on Binomial polynomials.
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within Pascal s Triangle that appear as different ways of organizing spacetime into wholes that appear as a set but are mass-like with their own continuities. The various concepts of the judgments and categories to do with quantity dance around wholeness. Schemas are varieties of wholeness that are implicit in the differentiation of Pascal s Triangle.But once Pascal s Triangle has differentiated then it is possible for a collapse back into one and the production of the field of Meta-anomalies in mathematics as the various side effects from that concrescence process. The dispersion of the separate anomalies are themselves seeds that are a diversity in the field of mathematics that are then a plurality that appears virtually. One of those anomalies are the differentiation of the Hyper Complex algebras that are in the virtual imaginary discontinuous space, and so the EMS cycle can be seen to start again in its quasi-causality.
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or Tetrahedrons. Gödel s proof is an EMS structure that spawns dissipative structures that in turn combine to give us images of the other Special Systems that turn out to be the polytopes shared by all dimensions. The third and fourth dimensions are anomalies and have more space in them and so they have more Platonic Solids than these unlike all other dimensions. Thus, the polytopes in the third dimension that are extra are a part of the Meta-anomaly. This means that the very thing that unground all other formal systems grounds the Special Systems, and based on that it serves as a ground for Schemas Theory, which in turn is a field that gives a bedrock to Systems Science, and Systems Science is the basis for Systems Engineering. Schemas theory is also the basis of Software engineering. Just like we were surprised that computational science was related to co-algebras when the duality of algebras and co-algebras we discovered so to this duality in the modalities of using Gödel s Proof to construct rather than to contaminate numbers by coding in paradoxes is unexpected. It is unexpected that what ungrounds all other formal systems grounds Special Systems Theory. And this makes Special Systems Theory a unique theory with what might be called a groundless ground. It is afoundational rather than foundational or anti-foundational. Afoundational means that the Proof is used for a different purpose to give rise to the Special Systems in the form that is produced by the Pascal Triangle and Tetrahedron that appears in all dimensions, those are the very dimensions that are spanned by schemas theory that gives us the context for understanding Special Systems Theory.
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in as much as they make possible and explain the physical and living and social phenomena that they describe so well. But beyond that we find that these special systems have a nondual nature and that the mathematics itself defines that nonduality by multiple analogies and so it seems that the mathematics itself is ordered in such a way to set up norms to be broken by the mathematical anomalies and these help us understand and describe anomalous phenomena that seemingly over-reach the bounds of physical phenomena as well, but of course we find that similar anomalies exist in physics. And so, we come to know that the nonduality that some forms of mathematics describes is intrinsic to the math itself, and not merely an interpretation, rather it is telling us something about the structure of reality that otherwise we would not know if those mathematical analogies did not exist. Because the Special Systems are describing phenomena like Life, Conscious and the Social it is describing phenomena delved into by (usserl s Phenomenology, especially as seen in Genetic Phenomenology33 and as augmented by Gurwitsch34 and Merleau-Ponty35 as
well as Deleuze 36 . Special Systems describes the constraints under which
Phenomenology operates and exists. But these constraints are not obvious in the phenomenology of Husserl because he takes their anomalous and rare properties for granted and does not search for the source of those properties. But in the development of Phenomenology under Gurwitsch, Merleau-Ponty and the Transcendental Empiricism of Deleuze these constraints become clearer and we eventually see that the constraints are produced by the Special Systems and especially the special systems as the medium for the interaction of the four temporal dimensions of Emergent Time. But what we see is that we have to think outside of the box of Gödel s Incompleteness Proof in order to see the importance of Mathematics and Logic as defining the possibility of the proof which in turn grounds the Special Systems which in turn gives the constraints for phenomenology as understood through the lens of Transcendental Empiricism. So, it is a cycle. Mathematical Logic defines the possibility of the Special System but then Special Systems grounds mathematical logic in a virtuous cycle that is afoundational. What ungrounds everything else ground Special Systems, that in turn ground Schemas Theory that in turn ground mathematics and logic. Mathematicians consider Gödel s proof tangential as it does not destroy anything essential but remains notional as an indication of a limit and the ultimate groundlessness of Mathematics. But there is a very narrow grounding of Special Systems which is a bit like the series of integers to infinity being equal to -1/12. It is something inexplicable. But once we have Special Systems grounded then we can ground schemas theory, and by that the rest of
33Welton, Donn. Other Husserl: The Horizons of Transcendental Phenomenology. Bloomington, Ind: Indiana U.P,
2002.
34Gurwitsch, Alan. The Field of Consciousness. Pittsburgh, Pa: Duquesne Univ. Press, 1964.
35Merleau-Ponty, Maurice. Phenomenology of Perception. Princeton, N.J: Recording for the Blind & Dyslexic,
2007. Merleau-Ponty, Maurice, Claude Lefort, and Alphonso Lingis. The Visible and the Invisible: Followed by Working Notes. Evanston [Illinois] : Northwestern University Press, 2000. Low, Douglas B. Merleau-ponty's Last Vision: A Proposal for the Completion of the Visible and the Invisible. Evanston, Ill: Northwestern University Press, 2000.
36Deleuze, Gilles. Difference and Repetition. London: Bloomsbury, 2014. Deleuze, Gilles. Logic of Sense. Place
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mathematics because Schemas Theory produces the countable things and the continuities that we recognize and differentiate and then abstract from to produce mathematics as described by the Kantian Meta-episteme. Grounding happens in a roundabout way. It happens through the mechanism of ungrounding turned to another purpose. It is not mathematics that is grounded per se but the Special Systems and then indirectly the Schemas and then even more indirectly mathematics of various kinds through their anomalies association with the special systems. So we see that Kurt Gödel was right. His Theorem showed that no formal system of any worthwhile complexity, such as Set Theory or Arithmetic, was liable to contamination of Paradox despite the precautions of Russell s Ramified Higher Logical Type Theory. We can code into the numbers theoretical statements that result in paradox and derive the paradox from them within the numbers. So paradox intentionally set as if by hacking cannot be avoided. The security system of Ramified Higher Logical type theory is not enough to avoid paradoxical contamination. But oddly the very structure by which we prove this has the structure of the Emergent Meta-system and we can produce Dissipative Ordering Special Systems with the same resources we prove contamination using co-recursion and Peano s Arithmetic. If we conjunct these dissipative structures, we can construct Autopoietic Symbiotic and Reflexive Social Special Systems. But the images of these are just the Simplicies and the Cross/Cubic polytopes in each dimension. These are direct images of the various Special Systems. Once we have grounded the Special Systems in the Gödel Proof then we can go on to ground Schemas Theory so we have something countable and then Mathematics as the orderings of the countable things or their shapes. And this indirect grounding also gives us a basis for our model of the Worldview based on meta-dimensions and fibered rational knots. In other words, it is the indirection that no one was expecting and also it was the fact that it is not mathematics that is grounded by Special Systems which is an image of the Meta-anomaly as a field of sparse, rare anomalies spread out through mathematics that has a family resemblance to each other. Not what we wanted to find but better than no grounds at all. But it means we have to settle for groundless grounds of the type that Heidegger talks about in Being and Time37.
Gödel s Incompleteness Theorem means we have to decide whether we want incompleteness or inconsistency in our formal models. We must have at least one. And that ultimately means that if one is absolute then the other is relative, and this introduces Process Being into mathematics which is mostly about Pure Being. And because there is a difference that makes a difference between Pure and Process Being that is necessary that introduces Hyper Being into mathematics as well, and the inverse of Hyper Being is Wild Being. So, the frontier of mathematics is seeing in it the various meta-levels of Being. And these various meta-levels of Being have been developed over the course of the development of Fundamental Ontology in Continental Philosophy. It turns out that the kinds of Being and the Special Systems interleave with each other and form a duality each series defining the other series. And of course, the meta-levels of Being is an example of the application of Ramified Higher Logical Type Theory to a specific concept: Being within the Western
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worldview which itself is an anomaly, as it appears in no other languages than those of Indo-European decent.
