Mathematical Meaning 01 On the Meta anom

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
kent@palmer.name
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

Mathematicians avoid assiduously any assignment of meaning to mathematical
structures. It is taboo to speculate about the meaning of the structures they invent or
discover. However, a Renaissance has occurred in Mathematics over the last hundred
years or so. There are so many new kinds of Mathematics and proofs of old problems
are becoming commonplace. It seems as if Mathematics is an ever expanding universe
within the Platonic Realm which like the physical universe is accelerating in its
expansion with no end in sight. But it has no meaning except that we give it. There are
myriad kinds of order ever more subtle. Occasionally we find something in physics
that is related to that order, and those isomorphisms are uncanny when they appear.
But there is no reason assign that order or another, there is merely a match that
makes physical calculations easier and emulates the physical phenomena for unknow
reason. Or perhaps the phenomena emulates the mathematics. It is unclear why the
isomorphism occurs. But when we discover it, then there is a eureka moment, and we
consider that a deep and genuine discovery. People get the Nobel prize for that kind
of discovery. The Clay prize is less prestigious, for there is no Nobel prize for
discoveries in Mathematics itself, only for applications of Mathematics. What we want
to consider here is the meaning of mathematics itself and how in part that might be
1

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 Void 1. In that book I explore the
structure of the Western worldview and toward the end of that book I decide to read
Plato s 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, Nonorientable surfaces, but also in some analogies in physics like solitons, superconductivity 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 Prigogine 5 and his
negative entropy6. Autopoietic Symbiotic Special Systems were associated with the
theory of Maturana and Varella7. Reflexive Social Special System was associated with
1 https://works.bepress.com/kent_palmer/2/
2 https://www.wikiwand.com/en/Laws_(dialogue)
3 https://works.bepress.com/kent_palmer/3/
4 Koestler, Arthur. Janus: A Summing Up. Nueva York: A Division of Random House,

1979. https://www.wikiwand.com/en/Holon_(philosophy)
5 Prigogine, Ilya, and Isabelle Stengers. Order Out of Chaos: Man's New Dialogue with Nature. London: Fontana,

1988.

6 https://www.wikiwand.com/en/Negentropy
7 Maturana, Humberto R, and Francisco J. Varela. The Tree of Knowledge: The Biological Roots of Human

Understanding. Boston [u.a.: Shambhala, 2008. Maturana, Romesin
́ H, and Francisco J. Varela. Autopoiesis and
Cognition: The Realization of the Living. Dordrecht: D. Reidel, 1980. Mingers, John. Self-producing Systems:
Implications and Applications of Autopoiesis. Springer-Verlag New York, 2013.

2

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
papers 12 . 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

8 O'Malley, John B. Sociology of Meaning. London: Human Context Books, 1973.
9 Sandywell, Barry. Reflexivity and the Crisis of Western Reason: Logological Investigations. Place of publication

not identified: Routledge, 2013.
10 Blum, Alan. Theorizing. London: Heinemann, 1974.
11 Klir, George J. Architecture of Systems Problem Solving. Place of publication not identified: Springer, 2012.
12 https://www.academia.edu/3795281/Special_Systems_Theory https://works.bepress.com/kent_palmer/4/
13 https://independent.academia.edu/KentPalmer/Emergent-Worlds:-Being,-Existence,-Manifestation
14 https://www.academia.edu/9868340/Exploring_the_Dagger_or_

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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
16 https://www.academia.edu/31038671/Foundations_of_Systems_Architecture_Design
17 Kurke, Geoffrey. Coins, Bodies, Games, and Gold: The Politics of Meaning in Archaic Greece. Chichester:

Princeton University Press, 1999.
18 Hahnemann, Samuel, and Constantine Hering. Organon of Homoeopathic Medicine. Charleston, S.C:

BiblioLife, 2010.
19 Ameke, Wilhelm, and R E. Dudgeon. History of Homeopathy: Its Origin, Its Conflicts. New Delhi: B. Jain Pub,

2007. Haller, John S. The History of American Homeopathy: The Academic Years, 1820-1935. New York [u.a.:
Pharmaceutical Products Press, 2005.

4

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 AntiOedipus 21 and Thousand Plateaus 22 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
21 Anti-oedipus : Capitalism and Schizophrenia. University of Minnesota Press, 1998.
22 Deleuze, Gilles, and Felix Guattari. A Thousand Plateaus. Minneapolis: University of Minnesota Press, 2007.
́
23 Deacon, Terrence W. Incomplete Nature: How Mind Emerged from Matter. New York: W.W. Norton & Co, 2013.

5

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

24 Fa, zang, and Dirck Vorenkamp. An English Translation of Fa-Tsang's "commentary on the Awakening of

Faith". Lewiston N.Y.: E. Mellen, 2004
25 Saunders, E D. Mudrā: A Study of the Symbolic Gestures in Japanese Buddhist Sculpture. Princeton, N.J:

Princeton University Press, 1985.
26 https://www.academia.edu/2223570/Category_theory_and_concrete_universals

6

Emergent Meta-system and was describing it in his Monadology 27 . 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 Metaanomaly 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
27 Leibniz, Gottfried W, and Robert Latta. The Monadology. Adelaide: University of Adelaide Library, 2008

7

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. NeoPlatonism 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_Metasystems_and_Holonomic_Special_Systems_Theory
29 Fuller, R.B, and E.J Applewhite. Synergetics: Explorations in the Geometry of Thinking. New York: Macmillan,

1975.
30 Manning, Henry P. Geometry of Four Dimensions. New York: Dover P, 1914

8

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.
What we are suggesting is that we help each other try to discover more intrinsic
meanings of mathematics from a philosophical perspective by studying further the
ramifications of Special Systems Theory and Schemas Theory as well as Dagger
Theory. I am sure that there are more types of mathematics that are nondual
analogies for Holons as well as more precursors out there, as well as more people in
the Western tradition that have rediscovered the Special Systems for themselves
independently. For instance, Jung s model of the self in terms of archetypes can be
seen as a representation of the Special Systems. Some of Lacan s pronouncements can
also be seen in this same light. Victor Frankl gives a model of the openly/closed
system that relies on Fourth Dimensional ways of thinking that there are more ways
to get into a sphere than piercing its surface. Koestler s definition of the Holon is
9

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.
Where does the Meta-anomaly come from? Daniel Smith in his book of essays on
Deleuze, one of the best commentaries I have read, mentions in one of the essays that
there are three different creation accounts: Creation internally (Spinoza),
Emanationist (Neo-Platonism), Creation externally which is the normal theory that
comes from the Bible. But there is a fourth possibility not mentioned by Smith which
is collapse from plurality to unity. If Emanationism was the last word then there
would be no reason to have the meta-anomaly. Mathematics would look like a pearl
with no quirks within its fabric. But there are anomalies in its structure like Aliquot
numbers, and this suggests that there is a catastrophic collapse from plurality to
unity, and in the process fusion occurs and strange properties and structures are
produced that are unexpected which we discover, like the Imaginary Numbers, or like
10

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 Badiou 31 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 deemergent 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
31 Badiou, 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 badly 32 .
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.
32 Freedman, David N. The Nine Commandments. New York: Bantam Doubleday, 2000.

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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 corecursion 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.
It is clear that this de-creation cycle that produces the meta-anomaly in mathematics
that is articulated in Special Systems Theory is a completely different way of thinking
about things than we are used to. We are used to starting with one in the natural
numbers and then going toward plurality and eventually complexity. However, we
have no explanation of how Oneness arises. We are not used to thinking how that
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Individual unit that is are starting point is itself produced. Badiou wants us to think
about that and he says that a miracle occurs that produces the ultra one out of the
multiple as unique event. But still he does not explain how that can happen, it is
something he merely assumes as the basis of his Set Theory ontology. He needs the
Multiple and the Event to make Set Theory into a plausible ontology, i.e. to give it
some dynamism. But this is a miracle that is unexplained. But when we look for
evidence of what really happened to produce the ultra-one then we find the anomalies
that in a field have similarities to each other we call the Meta-anomaly that are
oddities and peculiarities scattered around mathematics with similar structure in
each case, like a sparse dispersed fractal that goes throughout mathematics and gives
unexpected results that show that Mathematics is at its foundation discovered not
invented or constructed. Invention and Construction is a means of discovery. If it were
only constructed and invented the Meta-anomaly would not be there. Order would be
itself well ordered. Instead there are these odd properties that appear anomalously
as sparse dispersed phenomena that we find in our mathematical research that gives
finite answers in an infinite sea of possibilities. These dispersed anomalies that are
similar to each other are in fact coordinated and are part of the same higher order
structure within mathematics. A bit of moonshine of a different type that relates to
anomalous physical structures. They provide the conditions for the possibility of
certain rare phenomena. It is this Meta-anomaly that causes us to consider, contra
Badiou, that there is instead a ultra-plurality or assemblage that is produced first out
of the Multiple. The multiple is not complete heterogeneity and incommensurability
but is instead like a Rhizome as Deleuze and Guattari imagine. There are local pockets
of order and commensurability within the overall background of the Multiple that is
based on disorder and lack of similarity or resemblance. The Multiple is a nihilistic
background to the unity and totality of the ultra-one as a singularity. But not just one
ultra-one is produced rather it is one of many and thus an ultra-plurality comes first
before we focus on any one unit, and this assemblage collapses into the ultra-one. In
that collapse the field itself that is between the individual units of the plurality
becomes folded and discontinuities appear within that meta-systemic (openscape)
field. And those folds, fusions, and discontinuities become the different embodiments
of rare fusion structures of the Meta-anomaly. Out of many one is forged not without
difficulty that leads to peculiar and unique properties of many low cardinality or
ordinality numbers. When we take all these mathematical anomalies together we see
that they have a similar but varying structure depending on the kind of math they
appear within. We generalize across the anomalies based on their similarity and we
call this the basis of Special Systems. Once we have a theory of anomalous systems
then we look for examples and we find physical anomalies with a similar structure
like solitons, super-conductivity Cooper pairs, and Bose-Einstein condensates. There
are many anomalous properties of things in nature and we can look for those that are
similar to the mathematical anomalies and then use the mathematical anomalies to
explain their peculiar structures in physical existence that are unexpected. Special
Systems Theory is a generalization of anomalies from the structure of similar
mathematical and physical anomalies that we then use to attempt to explain other
important large scale anomalies that appear phenomenologically in many ontic
realms that appear in thresholds of emergence, 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 th