Download Word doc - Austega

COMPLEXITY AND LINEARITY
by Florian Colceag
The debate about using complex or linear models in sociology has a long history that
is closely related to several political pressures. All totalitarian regimes preferred linear
models. The main reasons for this preferences are: a good command chain involved by
these linear logics and models, simplicity leading to obedience, and graph extensions
leading to various versions of decisions in different contexts. As long as the world looked
unlimited in resources and horizons this approach was functional but unfair for the social
and natural environment. Now the world is much smaller, and will be smaller as an effect
of the population explosion, ecologic damage, exhaustion of the natural resources, and
new informational technologies. All these problems require complex approaches.
Linearity was very much used in national policies modeling. Complexity was usually
avoided because is difficult to use, gives uncontrollable predictions, and lead to free
thinking that might find ways to break old thinking patterns. This political problem
creates an undeclared opposition for complexity in sociologic modeling. Complexity
might be difficult to manipulate, but linearity hardly responds to modeling needs, creating
unpredictable tensions. An example is the apparent failure of several globalization
programs. Solution can be found in complexity area.
The instrument for complex analyses created by feedback cycles, algebraic fractals,
and cellular automata might be a solution for technical applications using computer
programs. The main problem is: is it an adequate instrument for sociological description?
The answer and justification for these instruments are the following:
-The mathematical background for these instruments is projective space. This means
the study of the relationships among various objects that doesn’t include measurements.
-Feedback cycles are described in two ways; two sets of objects that generate each
other, and two sets of relationships that generate each other. The way in which
relationships that contain transitory information can modify objects containing structural
information can also be described.
-This description can be done using a dynamic structure, cellular automata hypercubes created by using algebraic fractal’s theory.
-Algebraic fractals develop structures that are not identical, but change shape and
rules enriching their properties to superior levels, but preserving and respecting also rules
developed in previous levels. This aspect of complexity can be noticed from super strings
to particle physics, from particle physics to atoms, molecular physics and chemistry, from
chemistry to biochemistry life phenomena and from life to intelligence, social behavior
and sociology, etc. There is no evidence that the theory described here cover all this
complexity. It only describes a similar complexity in mathematical terms.
All these phenomena are closely related with relationships between various “objects”
specific for each level, and in all these domains projective space is used for modeling
reality. Relationships can be described by feedback cycles; behaviors can be described by
various cellular automata build using algebraic fractal’s theory. This theory can be
extended in algebraic varieties domain describing more complex mathematical patterns
for different levels of complexity.
1
For smaller targets as social modeling or life modeling feedback cycles theory in
conjunction with algebraic fractals theory can develop specific models on the projective
space describing specific characteristics. For example symmetry characteristic that can be
noticed in any of these domains can be described using the figure given by Pappus
theorem, where three feedback cycles are correlated to each other. (See the figure)
These characteristics are common for symmetric populations arrangements in
structural patterns (See Romania internal perspective), for the symmetric and complex
interrelation among endoderm, mesoderm, and exoderm in organic informational
relationship, and for many other domains including self-stimulating and self-inhibitory or
oscillatory systems described by Prigogine. In life processes self-stimulating and selfinhibitory systems are connected to each other creating feedback cycles able to regulate
informational organic complexity.
Cellular automata, created by using feedback cycles as component element, have also
an interesting behavior. These automata can connect to each other in large conglomerates,
are able to create informational residues that can develop other cellular automata, and are
able to compute information and to find decisional alternatives including Turing
machines on their faces.
Actual limits of this theory are created by the projective space, and by an insufficient
study of the theory. The theory is developing in various directions. By including on the
projective space a specific metric for each algebraic fractal level, the theory might create
the potential for predictions in time. Time axe is already included in the structure of
information given by feedback created using authomorphisms of the projective space.
These authomorphisms are not commutative for composition, the succession of events
being irreversible.
A very good theoretical background for this theory is not yet completely developed,
but using only the few instruments already created its applications may be relevant in
2
understanding complexity. Part of this background is already built. Catastrophe, chaos,
knots, Gabor equations and fractals theories may cover several aspects including metric
relationships on projective perspective of algebraic fractals. Other connections will
develop in time. The new complexity theory of fractal varieties is now in study. This
theory will extend algebraic fractals.
Feedback cycles on projective space describing objects’ structure can develop as part
of a specific graph theory. This part that will be more descriptive will be useful in the
determination of the place of various qualities obtained by practical studies. These
qualities can be transferred in relationships language by identification giving the
possibility for direct application using cellular automata in various domains. For example
considering social group relationships on three main positions as dominating, dominated
and pacifist we can transfer these characteristics to 120 degrees rotations, 60 degrees
rotation, and identity that has the same order relationship. Rotation was observed also in
social groups as well as emancipating tendencies. Symmetry can be associated with a
different behavior, creative imitation, the two inversions can be associated with the
tendency for infinity for the inversion that is usually in relationship with rotations, and
fullness-emptiness characteristics for the inversion that is associated in algebraic
feedback cycles with symmetry. Every social behavior can be superficially characterized
by using these qualities. Introducing specific characteristics in cellular automata that
characterize the pattern of the studied object we can predict its behavior. For a better
approximation and determination of behavior more characteristics has to be included,
each one correlated with feedback cycles and aggregated into a multidimensional cellular
automata.
Cellular automata can be built by using specific characteristics describing
dimensions of structuring for each side of the cubic or hyper-cubic structure. These
dimensions are characteristic for various studied objects, and described by feedback
cycles, and must correlate to each other in the design of the sides’ frame of the cubic
(hyper-cubic) structure. Faces and other components can be fixed, or dynamic holes will
move from a place to another, creating computational Turing machines. These
applications can be done using computer programs and may describe potential future
behaviors of the system represented by the cellular automata. The accuracy of predictions
will be directly dependant of the quality of information included in the informational
model described by feedback cycles. These predictions will not include time as long as
only projective space is used without a specific metric for the algebraic fractal level
included. Describing only relationships among components the system will give the
chance for corrections, or completion of the information and can lead to a new kind of
informational technology.
I cannot predict how this theory will evolve in physics. It is possible to connect with
super string theory by developing more algebraic fractals levels, and by including fractal
varieties theory. If this will happen will be evidence that informational feedback cycles
are really the minimum basic information that leads to the formation of the universe. At
this moment everything is at the beginning, and a lot of work is required for a decisive
answer. If this link will be created that will be an implicit proves that universal cellular
automata perspectives created by Von Newman and David’s Bohm philosophy about
determination are correct. The effort to find information in this direction may be long.
3
Feedback cycles theory is the exploitation of “the nothing”. Due to the fact that by
composing the authomorphisms of a feedback cycle will obtain one, these feedback
cycles remained unnoticed. In fact this is the reality of every object. Each object
interferes with other objects in a very limited way, preserving its individuality. We can
ignore the existence of an object that is not very close to us, but it can act in a dramatic
way if will interfere with us. This aspect may give correction for chaos and catastrophe
theories creating a better perspective about prediction potentials.
Algebraic fractals describing one kind of feedback cycles show how different
feedback cycles formed by authomorphisms of the projective space can form bigger units
replacing these authomorphisms and forming a different kind of feedback cycles more
complex and containing all the old information and new information. Old information is
also transformed from a structure (group) to a new structure more general (variety). The
mathematical procedure uses symmetry and non-symmetry in an organic texture, creating
a pattern of understanding of potential interferences of these two properties. It also
describes cycles and circuits with informational value that are not feedback cycles. All
these contexts can be used in description of complex behavior for cellular automata
describing various phenomena including life processes. Because the procedure of
complex determination of new feedback cycles is inductive we can obtain more complex
feedback cycles that can be expressed in multiple dimensional matrices. From the
modeling perspective this fact corresponds to the possibility of dividing components of a
feedback cycle in smaller feedback cycles, connect these smaller feedback cycles
component by component and translate information obtained by connection in specific
language corresponding to the model.
Using this apparent isolation two systems will interfere using a code of
communication and passwords. This aspect is described by algebraic fractals when two
feedback cycles are composed with each other. Only several characteristics containing
these “passwords” are preserved in the product.
Algebraic fractals describe also a different aspect of the theory. A feedback cycle
formed by authomorphisms is obtained by composing different authomorphisms in
different ways. Some of them are composed from left to right, other are composed from
right to left. This interesting behavior is preserved into the next level and suggest the
tendency for return to origins, to check the information if fits or not with the previous
pattern. These characteristics create also a link with graph theory in which each point of
the feedback cycle has a functional value as data entrance, processing entered data, using
entered data, exit data, processing exit data and using exit data. This particular link may
be the key of understanding of many specific reactions regarding living systems that can
be characterized as complex cellular automata.
A different aspect of the algebraic fractal theory is spin behavior. This characteristic
can be correlated with every domain from physics to biology. It proves the existence of
several structural invariants that may correlate well with fields (even biologic fields)
convergence criteria of information in organized structures and other aspects.
The graph of composition for authomorphisms that will be respected by any other
algebraic fractal level is similar with strings composition. This fact also suggests a
possible complex approach towards string’s theory that might be searched. These
suggestions might lead to nothing or to the discovery of new rules. For the moment more
in depth studies does not confirm all these directions. The potential for a universal theory
4
described by including feedback cycles will remain a task for future research. At this
moment my attention is focused on sociological application mostly because of the
contemporary crises that need to be solved. New methodologies using computer
applications and a better start point has to be developed and because humanity has as a
main characteristic structures based on communication, I consider that feedback cycles
on algebraic fractal theory to be a potential useful instrument.
Any theory has its limits. A good theory will explain only the part of reality that is
targeted by its instruments. This theory can describe only relationships among
informational cellular automata, describe informational metabolism, and find out
potentials for change including growing up potential, or degenerating potentials. This
theory is not a limitative theory on fixed structures, doesn’t create an image about a fixed
patterned world, by contrary gives the image of a very interactive and dynamic world.
Only future practical studies will be able to prove the accuracy of this theory and its
utility. Computer programs simulators using various algebraic fractal levels will be
required to prove it. It is a new domain that needs confidence to develop, but it might be a
potential solution for the complex problems that challenge humanity in this time.
Complexity starts from a complex mathematical perspective. Feedback cycles theory
profit in the first stage by self-determining systems of objects in two sets able to generate
each other. The first set will generate the second set and the second set will generate the
first set. These properties we find first in Euclid axioms regarding points and lines, but
we can find similar phenomena in various contexts. Any time when these phenomena
appear they order the space in ordered structures on various levels of complexity. This
leads directly to algebraic varieties language for fractal varieties.
For example if we take as supplementary hypotheses that three circles generate the
same point (true for concurrent circles) and taking circles with equal radiuses, we obtain
an interesting phenomenon. Taking A, B, and C three circles that intersect each other in
the same point P, but intersect two by two in points R, S, and Q, than R, S, Q will
generate another circle with equal radius. In this case, three circles generate one point (P),
and three points will generate one circle (R, S, Q). Let’s call the fourth circle D. We have
now a phenomena able to order the plan through this new perspective of other two sets of
generators different from points and lines, this time will be points and circles. We can
describe by this procedure a different space izomorphous with the plan generated by
points and lines obtained by applying a transformation; in our case inversion applied on
the three vertexes and orthocentrum of a triangle (See Algebraic Fractals-Fractal
Varieties). This structure will be in isomorphism with other structures obtained through
the same procedure of self generating phenomena, obtaining through this procedure two
things: one fractal using algebraic structures in which we have isomorphism instead of
identity, a structure that mathematically can be described as variety where at certain
levels various phenomena can be linked together. For example three circles generating
one point and three points generating one circle is a relationship similar with three plans
generating one point three points generating one plan. This similarity will function as
long as we preserve the intrinsic conditions for both problems. Circles must be arranged
using a special procedure, and not any three circles will determine one point. These
details suggest a fractal development of larger phenomena in which several mathematical
objects are included: domains including referential systems, relationships among objects
like generating relationships describing feedback cycles, transformations and invariants
5
for transformations, geodesics and fields of solutions, and meta-structures (See Algebraic
fractals-fractal varieties).
Interpreting these results we can see how information in living systems can order
through a special ordering direction. Evolutionary theories don’t explain how complex
organs like eye or ear can form. This algebraic fractal perspective suggest that eye
complexity is due to its functionality correlated with an intrinsic developmental pattern
that respects several functions that is common in every developed object including living
objects.
This phenomenon is also visible in chemistry and was described by Prigogine, in
cosmology and physics. Prigogine described oscillatory reactions and chemical clocks
where all molecules participating in the chemical reaction act in the same way at the
same time. In cosmology it is also visible the structuring way for various cosmic objects
at various levels. Social sciences studied mass phenomena in which by creating a social
dipole large populations of humans or animals act in the same way. Biology can see this
phenomenon of coordination happening in every phenomenon. The transfer from
mathematics (just few examples) to generalized phenomena is hazardous, but has a logic
that can be followed by various scientists. The logic is the following one: Selforganization systems (feedback systems) can be noticed when we work with axioms,
geometric objects, authomorphisms (See cellular automata, algebraic fractals), varieties
(document in work). This large spectrum of theoretical objects fitting with this pattern is
also very visible in applied sciences. Feedback phenomena can be seen in absolutely any
applied sciences domain, but is not usually described using six necessary steps.
The variable number of steps describing feedback phenomena can be explained by a
property of higher-level feedback cycles structure. These structures can be
mathematically described in two ways; hyper-cubic matrices, or fractal spiral that will
give a linear perspective when we fix the number of stages of development for the fractal.
Hyper-cubic matrices will describe complex feedback cycles using dimensional
approaches. Spiral fractal presentation will describe complex feedback cycles using a
finite number of coordinates. The hyper-cubic matrices are obtained by differentiating a
six steps feedback cycle, each vertex of the feedback cycle being a new feedback cycle
with new characteristics. By correlating data entrances from each smaller feedback cycle
we obtain a new kind of cycle, totally we will obtain six new cycles that are in dependent
correlation with the firs set of cycles obtained by the differentiation of vertexes of the
main cycle. (See cellular automata algebraic fractals). This kind of phenomena is visible
also in various mathematical domains. An example in the elementary axiomatic geometry
is the following one:
Take two directly similar triangles ABC, and A’B’C’ and build three other similar
triangle to each other but not necessarily with the first two AA’A”, BB’B”, and CC’C”.
You will obtain a new triangle A”B”C” that will be similar with ABC and A’B’C’.
We have in this example two sets of triangles that generate each other in a
complementary way. Because triangle structure can be described as a feedback cycle
generated by points and lines the pattern of this structure enter in a complex two
dimensional feedback cycle structure. It is also interesting that introducing this kind of
self-generating objects the entire plane will be covered in a continuous way with
elements of this structure. Taking points with the same baricentric coordinates in one set
of triangles we obtain triangles from the second set having these points as vertexes.
6
The same phenomena was found in algebraic fractals describing a new level of
complexity in the structure of feedbacks created using projective space authomorphisms
for the particular group presented in the paper (see cellular automata algebraic fractals).
Similar phenomena are common in algebraic varieties, where varieties and module spaces
correlate to each other, and in many other mathematical domains. The phenomena may
be difficult to determine when the set of generators create a continuous phenomenon, not
a discrete one. This is the main reason for which this phenomenon was not noticed
before.
Fractal spiral version can be created by differentiating a spiral with six loops in six
new smaller loops for a big loop and so on. Each differentiation will create a new set of
smaller loops for the previous set of loops. We can describe this fractal using a system of
coordinates. Fixing n-1 coordinates and considering the other coordinate we will obtain
cycle describing a dimension (direction) of structuring the information in the spiral. If we
will stop to obtain new loops we will have a complex description: using a linear
connection of information we will consider the spiral in isomorphism with a line and
describing everything in a sequential way. Using the system of coordinates we will obtain
connections at information levels. Even if these two mathematical approaches are similar
they reveal different phenomena.
Applying these fractal procedures in biology we open a new set of questions
regarding cell’s structuring, DNA structuring and information, the organism coordination,
etc. We don’t have enough information to describe these cycles in biology mostly
because without a research philosophy and logic envisioning complexity obtained results
are not structured. DNA structures like genes and chromosomes are three-dimensional
structures fold in a complex way. This complexity was not assessed until now,
reductionism fixing only the linear perspective. It is very possible that following this new
line of thinking in which the spatial structure be considered to obtain new information.
The relationship among DNA sequences and synthesized proteins was regarded only
from DNA to proteins, disconsidering the other sense from proteins to DNA. Chemical
structures were considered as an objects base for the structured information in organic
structures without considering potential biological fields mathematically described by
authomorphisms or functions. An attempt in this direction is Gabor equations considering
the network of information at the cerebral level. It is very possible that following the new
line of research created by complexity and fractal theories to understand later life
behavior in its entire complexity. The mathematical tools for this research are only in the
initial phase but can develop in time.
The importance of this development and change of direction in complex research is
extraordinary and very urgent. Human’s species relationship with the environment,
globalization and new technologies able to protect the environment and to assure social
stability, prosperity and progress in the same time need these kinds of approaches.
Decisional policies and leadership has to be more scientifically determined in order to
eliminate the permanent risk of disaster created by the actual forms of leadership. We are
in front of an enormous challenge, we don’t have enough tools and instruments to
manage it, but we have a chance to develop what we need with great effort.
Algebraic fractals theory develops also a new perspective about cellular automata.
We can classify more than two kinds of cellular automata using this theory. The first kind
describes simple cellular automata in a cubic or hyper-cubic structure, where sides are
7
feedback cycles. All other components try to develop feedback cycles in any direction,
but imposed conditions for these cycles create contradictory potential solutions for cycles
that following different directions will intersect to each other. This means that if one
cycle will become a feedback cycle will destroy another concurrent cycle that was a
feedback cycle before creating an error in the old one. These errors will move from a
cycle to a different cycle in the hyper-cubic structure, creating chaos of organization that
will create several patterns of behavior in a metabolic way. Any information introduced
in these automata by modifying one component will be digested by it. If an automata will
not be able to digest the information will link to a new automata. Using algebraic fractals
created with authomorphisms of the projective space as elements for feedback cycles we
will obtain also Turing machines for any knot of the automata, able to decide how this
automata will act.
Cellular automata can be also described by increasing the number of dimensions of
the matrix, by differentiating each component into a feedback cycle. This procedure can
give information about the internal logic for choosing one variant of decision instead of a
different variant, and better define internal Turing machines.
A different kind of automata will be opened automata. By using transformations
various feedback cycles will be transported in feedback cycles of a different kind, or in
feedback structures. For example polar transformation transform lines in points and
points in lines. Taking a triangle there are three conics that associated with polar
transformation will transform each vertex in the opposite side and each side in the
opposite vertex. We obtain using this procedure two new objects that form a similar
structure as triangular structure. This new objects will be conics and four points. Any two
conics will intersect in the complex projective plan in a four-point, each two four points
will generate a conic.
The general procedure of transportation of feedback cycles structure will determine
applied to automata of the first kind a transportation in other automata in a different
domain with new properties added by characteristics of this new domain. We can define
by using this procedure, a second order automata, that is structured across several
domains and associated structures. Because this kind of automata uses both objects
(Triangular or other structures composed by triangular structures) and informational
vectors (transformations), we obtain a new kind of dynamic structure able to change its
objects as a result of new information, describing an evolving perspective.
From the modeling perspective this property will be able to permit correlations
among various connected phenomena in a similar way as chaos theory do, but correlating
information from various sources and interconnecting them. It will also give a new
perspective about evolution that seems to be oriented following several describable
patterns, and correlating various evolving objects that are in relationship with each other.
These initial principles able to organize information using computer application
programs are more a scientific philosophy right now than a scientific well determined
result. Future researches will confirm this philosophy. Even if there are not enough
evidences, different problems arising from biological area suggest that this approach is
correct. Cell organization, tissues or organs and organisms’ organization follow several
common patterns all responding to feedback cycles. There is certainly an organizational
pattern common for all these formations encrypted in genes. This pattern has to be a
natural law respected by chemical substances, and this reasoning can continue.
8
Starting from the hypotheses that the cycle of six elements will contain a filter of
selection for information called data entrance (A1), that limits selecting data by applying
initial conditions; data that are not selected are preserved in a different site (A2). Will
contain also one unit called processing entered data unit (B1), that will take this
information and will process following several already known algorithms; data that can
not be processed because there is no algorithm for processing it will be selected in a
different site (B2). It will also contain a unit called using data unit (C1) that will compare
the obtained results with results included in the structure of the feedback cycle,
preserving similar or recognizable results and selecting different results in a new site
(C2). Another site will be exit data site (D1) that will contain data that can be expressed
in a unitary style conforming to the stile of expression of the cycle (for example formula
used in generating the feedback cycle is a stile); if several data will not be expressible in
the known style will be selected in a different site (D2). Processing exit data site (E1)
will contain algorithms of comparing various data, selection of structures of data and
assessing non-contradiction and consistency of data; data that will not be assessable will
be selected in a different site (E2). The last site is using exit data (F1) where assessed
data from the previous site will be included in a generalizing or limiting algorithms; data
that cannot be generalized will be selected in a different site (F2). Generalized or selected
data will include all steps and will transform the first filter of data selection, now a
different set of data from A2 being included together wit A1 data. The new cycle will
restart this time considering element from rejected data in sites with coordinate 2. If the
cycle is stable sites with coordinate 2 will be empty, if the cycle will evolve, sites with
coordinate 2 will contain data. If the cycle will degenerate, cycles with coordinate 1 will
have fewer elements for any repeating sequence of steps.
This algorithm seems to be universal. It can be described in mathematics, but it can
be also described in physics, chemistry, biology and social sciences. This result can be
used even if the mathematical part is not yet completely developed, and will help in a
better understanding of various phenomena.
Feedback cycles can associate in several ways: by connecting in a linear way with
each other, by connecting generators with other generators, or by interconnecting in a
more complex way (See the figure). This result can be also universal.
Another way to associate feedback structure is described by simplexes configuration.
In this case more that two elements from one set will contribute for generating an element
from the second set (For example three circles generating the same point, three points
generating the same circle). These more complex feedback cycles can be directly
decomposed in simple feedback cycles. An example is tetrahedral structures that can be
decomposed in four triangular structures connected to each other (See algebraic fractalsfractal varieties). This is a different kind of feedback cycles connection that is not
described in the figure due to its complexity. The lat way to connect feedback cycles will
also correspond to new dimensions in characterizing complex structures that contain
these kinds of structures. New information that is not digested by the old feedback
structure will connect to each component of the old feedback cycle generating one more
dimension in complexity. Technical difficulties in following these connections require
computer programs able to model these phenomena and to assess different results.
9
Each way of connecting feedback cycles to each other will describe different facts
and can be associated with various phenomena. Third, fourth, and fifth examples show
how these feedback cycles will connect in networks. The sixth example shows how a set
of feedback cycles will connect in a second order feedback cycle in a fractal way that can
be also described in a two dimensional table.
Even if the mathematical description of algebraic fractals and fractal varieties is not
easy and not complete these connections can be independently used for modeling
networks and informational processes on these networks, describing in the same time
networks’ structures, and information structure, networks behavior (using cellular
automata), and creating the possibility for predictions of future behavior. For all these
applications computer programs are required.
Complexity can be described at least reporting to phenomena that show relationships
among objects. Each projective space described by using fractal varieties can be extended
on an afine space by introducing a specific metric. This will create a linear perspective
about the respective afine space that will be included in a new kind of complexity.
Florian Colceag
References at http://austega.com/florin/
10