REVIEW
OF THE SEMINAR UNDERSTANDING COMPLEXITY:
SYSTEMS, EMERGENCE AND EVOLUTION. CASE STUDY: THE CITY.
By
Verónica Coca Zancajo, participant as attendee.
The seminar is part of the PRIMER initiative for promoting
interdisciplinary methodologies in education and research, and it was held in
Leon (Spain) during the days 27th-30th of January of
2014.
The lectures were imparted by the professors:
Rainer Zimmerman, from the Munich University of Applied Sciences and Clare Hall of
Cambridge;
Gordana Dodig Crnkovic, from the Swedish Mälardalen University;
José María Díaz Nafría, from Hochschule München;
Paz Benito, from Universidad de León.
The topic of the
seminar is the Theory of systems applied to understand how complexity emerge,
structure itself and evolve; also the relationship between ethics and beautiful
according to the principles of urban-planning.
In order to
understand the relationship between urban-planning and the theory of systems,
we should analyse four main aspects of cities as open, complex and with a
hierarchical structure systems:
1. the urban structure, studied by
the classical urban planning, focus in the growing process, the physical and
social dimensions and the economic functions;
2. the urban processes directed
from public authorities, to the governance and management;
3. the dynamics of land uses, in
processes of expansion or intensification of use;
4. the internal flows (mobility,
economic transactions, transports, new information technologies, etc.) and
external flows (connectivity with another cities, the quality of interurban
spaces, etc.).
Introduction to dynamical systems theory
To begin this
study we must have a look back to the origin of the computational methods. Nowadays
Ramon Llull can be considerate the father
of computing and information science. By using the aristotelian logic Llull
posed the principle of computing
versus discussing, and he proposed
two main ideas: the knowledge as a tree of different science and the
combinatory logical machine able of verifying any sentence. The tree of
knowledge represents the idea of many different branches of knowledge that get
together to one unique trunk that is reality. So there are many different ways
to approach the knowledge of reality, but they cannot be in conflict. Llull
anticipate the mechanical calculation by symbolic representations, the
heuristic methods of artificial intelligence, the study of graph theory and the
semantic networks.
This vision of a
unique reality that can be known causes the determinism some centuries later.
After the Laws of Newton that allow
the anticipation of the position of planets (positivism), Pierre-Simon Laplace imagined the existence of a
demon, with no supernatural abilities but with the capability of specify the
exact position of any particle of the universe in every moment, so this demon
could know the past as well as the future. That idea was called determinism and
became a problem of free will.
However the
Newton’s equations were no able to solve the question of the interaction of a
third body in the position and trajectory of celestial bodies. This was known as the Problem of the Third
Body. To solve this problem a new theory was born: the K.A.M. theory.
In the twentieth
century many theories changed the idea of a unique reality that can be
determinate in every moment. Gödel
demonstrated with mathematical expressions that there are axioms and
formulations that cannot be verified inside their own logical system. So the
Mathematics are incomplete. Also the uncertainty principle of Heisenberg establishes the
impossibility of certain and precision in the observation of the movement and
position of a particle. The most we know the velocity of a particle, the less
we can determine its precise position. So after Heisenberg the tree of
knowledge is not unique any more, there are many possible trees of reality.
That is the reason why the theory of complex systems is being used in the new
researches in different fields of knowledge.
Reality versus modality
Reality itself
cannot be known since the appearance of quantum physics. So we are going to
take the ideas of Spinoza about
substance, attributes and modes. Reality is the substance, because it exists by
itself, so Nature is the substance. The attributes are infinite, for example the
extension of the matter or the thoughts (the mind), but they settle on modes.
Modes are physical or ideal objects. Instead of reality, we will talk about modality,
i.e., a representation of reality. This representation is not the same that the
reality represented. The modality is the representation of some qualities that
we, human beings, can perceive. Those qualities are just a part of the reality,
and we must have this in mind in order to evaluate the achievements of Science.
These representations are simplifications of a much more complex unattainable
reality.
Systems are
modalities created by human beings in order to understand Nature. Every system
has three inner qualities: composition (collection of components or agents,
matter or ideal), structure (links and relations among the agents, i.e., interaction
and organization)
and mechanism (processes of transformation inside the system, notice that
transformation needs energy). Every system has a purpose
that has to be accomplished in order to maintain the existence of the system. There
is an outer quality: the environment (things out of the composition, if they
interact with the components, the system is open, if not, it is close). The
systems can contain sub-systems and
be part of super-systems. These
modalities can be applied in physical science and in social science. The
holistic vision of the reality as a set of systems is called systemism.
Continuous model by Keller and Segel
Besides the
conception of systems as a set of transformations in the structure, used in
linguistics by Saussure and in epistemology by Jean Piaget (they both used
structure instead system many times), there is a mathematical definition of
system that includes the concepts of domain
and range. The dynamical system describes the range of change (transformations)
produced in the position in a very small portion of time:
dx/dt = f(x, t) when x = (x1,
x2, …, xn)
The analyse of
this expression can get high levels of complexity, when the functions cannot be
solved because they are non-linear and the solutions are infinite. That is the
reason why we will study the stability, by studying the critical points of the
system instead of the transformations. The expression to study the critical
points takes this form:
f(x, t) = 0
The points where
the function is equal to zero can be referred as bifurcation points, because the possible evolution of the system
can take different signs. To understand better this idea of bifurcation points,
a review of the dissipative structures
theory by Ilya Prigogine is
required. This theory introduces the non-equilibrium
and the asymmetry as main concepts
to understand the universe in opposition to the classical view. In the
nineteenth century Clausius
introduced the concept of entropy in order to express the grade of disorder in
a system. He formulated the second law of thermodynamics, usually expressed by
dS>0, that means: the entropy always increases inside an isolated system
until the non-equilibriums become equal, i.e., the Universe is evolving in
direction to the maximum disorder and this process is irreversible. Prigogine modified the second law of thermodynamics
in a way only applicable for open systems. The global entropy is the result of
this addition:
dS= dFS
+ dIS;
dIS
is the level of entropy inside the system, that is always increasing (positive);
dFS
is the level of entropy that flows out the system, so it is decreasing inside
the system (negative).
If the flow of
entropy out of the system is higher than the inner entropy, the global disorder
inside the system is lower; hence a new order is emerging. This process is
natural, produced by the system itself, so it receives the name of self-organization. This new point of view
for open systems has changed the bases of Science, because it provides us with
new mechanism to study the unstable sides of Nature -unfortunately Nature
cannot be describes appropriately by differential equations of dynamic systems,
because Nature has not the quality of continuity-. Prigogine also defeats the determinism, since new
structures can emerge spontaneously. Some of these processes have a cyclical behaviour; others appear in a random way (chaos theory). An example
of a cyclical process is the evolution of a colony of amoebae, studied by Keller and Segel.
The critical
points of the function, or bifurcation points as we saw, do not give us enough
information, in order to understand the context we need to study the
neighbourhood of the critical points. When we describe the whole system by a
critical point plus a very small variation, we call it linearization process. The critical points are unstable points
where the symmetry is broken and a new structure can emerge – innovation process -.
We will use a matrix
to express in a clearer way a process with many similar functions. Then one
matrix will be one state in the process of transformation. The initial state
(stable) can be the matrix E. Now we can express the transformations in terms
of negation and negation of negation:
E (stable state)
transforms into E*(unstable);
E* is the
negation of E: E*=N(E)
E*(unstable)
transforms into E** (stable);
E** is the
negation of negation of E: E**=N(E*)=NN(E)=N2(E)
We can represent
this transformations as a diagram of
evolution.
Networks and spaces
Talking about
systems we always must talk about space,
as a domain of influence or free paths for interactions. It is defined by
describing the boundary of the
system (where the rules or attributes of the space cease to exist, i.e., the
domain of influence is equal to zero). The space can be physical or virtual
(abstract). The geographical space, for example, is a projection of the social
space - in order to go into detail about the construction of social space it is
essential to read the works of Michel
Foucault as well as Henri Lefebvre -.
A Network is the dynamical core of
interactions, the transport of information. For example, a social network is
the connectedness through interpersonal relations.
The objects of
the system are always operating agents.
The agents can be human beings, groups, elementary particles, etc. The nature
of the agents does not change the result of the system.
Nowadays the
study of social networks has become very important to understand the changes in
the society, because of the complexity of these changes and because of the new
technologies. The studies are based in the computation of information for
social levels plus some intuitions for individual levels. The computation of
social information can be approached from two different perspectives: focus in human aspects or focus in computational aspects. This last
approach will try to create models of automatize calculations (based originally
on Hilberg program in 1900 and Turing Machine in 1936) for a better
understanding of the mechanisms that cannot be perceived in a simple view. Computing
social networks, we see that human groups are information processing networks,
but these groups are also information generators (human groups are sub-systems
and self-organized). Besides we can see that every system is always
self-referenced, because there is always a reaction to being observed, and also
every interpretation of another system is a bit of self-interpretation. Information
processing is a physical process of morphological change in the informational
structure, where the information is always relative to the observer. For human
connection as a multi-agent system we create this relation:
information>computation>cognition>information
For the
complexity of these social networks a new field of research was born: dynamic
networks analyses (see the work of Lazslo
Barabási). Looking for patterns of behaviour in complex social networks, a
fractal pattern has been found, and it makes possible the existence of a unique
model of growing independently of the scale (scale-free networks and self-similarity
as a quality of complex networks).
We will define a
category as a class of objects and a class of representation or morphism: the relation among agents of a
system under condition of every interaction has a unique agent-source and a
unique agent-target. When we represent the agents as a node or vertex, we can
talk about isomorphism.
We use networks
to represent the transport of information among the agents of the system. The
transport of information attaches the concepts of time and space (motion), but
we will construct an abstraction just to clarify the main structure of
interactions inside the system.
We will take the
Graph Theory of Euler as base line
for the abstract representation of the structure of the system (the
relationships among agents and its interaction) and use this network of vertex
and links as a formalization that allows us to provide different meanings to
the same structure of vertex and links. This theory is connected with topology. Topoi refers to location, but in Mathematical sense, topos is a
category plus an additional structure that allow as to operate “inside” that
“space”.
Besides
information, the system needs energy to make the transformation. Applying the
system to different models we can see than the role of the agents or nodes can
be very different form one model to another. When the agents wait until the
action of an external interaction happens, then those agents are passive and the representation is
mapping potentialities - classical
geographic map -. When the interactions happen among internal agents, these
agents are active and the
representation is mapping actualities - dynamic
processes -. See more about graphs and networks studying the model of the small world and isomorphic matrix.
Stochastic Petri Nets
Petri nets are another
kind of networks that allow us to discern the differences between objects and
morphisms. Objects will be named now species
(represented by circles) and morphisms will be transitions (represented by squares). The relation between boxes
will be define by vector (see Hamiltonian
function).These networks are active always and they show one benefit: we see
directly the result of the transformation as a new specie. So the Petri nets
are represented by input boxes and output boxes. The interaction can take place
among different species and with different transitions or changes of state.
Because of this, we define Petri nets as a set of species and a set of
transitions, with the functions that connect them. When the model is not
deterministic and there is some uncertainty about the result, then we said that
the network is stochastic, and we
should take in consideration the probabilities
of certain result or another. By iterations of the probabilistic calculation,
we can have very close results in very different fields.
We can represent
a matrix of all the spaces in a subspace. The entrances will represent the
interaction between species, zero for no interaction and one for interaction.
That matrix (called Hamiltonian matrix) will represent all the probabilities of
interactions and its expression is quite near the Schrödinger equation.
And important
point to define the processes of transformation is their character of
reversible or irreversible. A network is weakly
reversible if the transformation in one direction, from A to B, is also
possible in the way back, from B we can get A.
For these Petri
networks, the existence of an equilibrium solution depends on their character
of weakly reversible or not weakly reversible, and so their grade of deficiency - positive or zero - (deficiency is the number of vertex minus the
number of connected objects minus the order of the stoichiometric subspace).
In occasions the solution depends on time.
Applications of systems theory on urban space
The relation
between cities and networks seems to be evident nowadays, but when did the
urban planning meet the system theory at the first time? Thanks to Jane Jacobs and her book Death and Life of Great American Cities,
the classical urban-planning practically collapse and a new point of view was
required. This change has two important consequences: in one hand, the relation
between the complexity of the city and the complexity of natural processes was
born (the idea of ecosystem was applied for urban spaces), and in the other
hand, the planning process had to be inverted, becoming a processes bottom-up instead of top-down (democratization in the
decision process, citizens participation, etc.). The professor Bettencourt from Santa Fe Institute
notices that Jacobs refers to the city problems as organized complexity, and nowadays the challenge is still the
creation of a new urban –planning, more scientific and with better complex adaptive models. In that sense, Kevin Lynch contemplated in the ‘80s a
very important question: is the city an analogous model to other natural systems?
It is very interesting that the planning critics has placed a value on
vernacular architecture and spontaneously born settlements as slums, set against
the modern big designs of nineteenth and twentieth century. This consideration
entails two ideas: first, the emergent aspects of urban space when official planning
is missing and second, the importance of the transformation scale in urban
space.
Bettencourt has
established two main measurable properties: population
density and the use of urban material
infrastructures. He also considers the distance travelled, the
transportation of goods and people, etc. , and has made the comparison between
cities and stars in terms of attraction nodes, dissipation, etc. This metaphor is
valid for some properties of urban space, such as luminosity-activity, but it
cannot be a urban model for general studies, because many characteristic magnitudes
of stars are not valid for cities, for example gravitation. In order to maintain
this metaphor, we need to find how to change those magnitudes to others that we
could use for cities.
Fractality of cities
Michael
Batty and his partners were the first to publish a
work identifying the self-organized structure of British coast with the Koch
curve. They demonstrated the fractal structure across different scales for
cities.
Fractality is
present in the structure of quartiers, districts, etc. The classical example of
fractal space is the model of roman encampment that was extended to the new
colonies. The case of Bolonia is good example.
Hodological space
Hodos means routs or ways.
Taking the ideas of the school of Gestalt, we can create a representation of
the hodological space of neighbours.
This hodological space will be the living space of every individual,
recognizing morphological spaces in the personal routs, with clearly a subjective
connotation by the association of locations and feelings. This hodological
space put the geographical space in a secondary order. The map of living space will be the addition of
geographical space plus individual connotations. The superposition of the
hodological spaces of all the inhabitants will result the social hodological
space that can contribute to complete the other objective data analyses.
Final remaks
The observation of
the reality cannot be separated of our human capacities.
Modelling the
reality is re-creating the result of our observation, including processes of
interpolation and extrapolation.
Models are set
in theories. Theories are used to understand what is not human itself.
The whole
knowledge is just a human perception of reality, a human modality.
The concepts of
space (time), networks and systems are characteristics of human knowledge.
Those three
concepts are related to another triade, cognition, communication and
cooperation.
The ancient idea
of harmony posed the relation between adequate form and adequate content. So
harmony means the sum of ethics and aesthetics, resumed in the word kalokagathía.
Harmonic systems
according to Heraclitus are constantly in motion, always stirring the mixture,
always in conflict.
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