Untangling complex syste.., p.7
Untangling Complex Systems, page 7
of-equilibrium systems having emergent properties are examples of what we nowadays refer to as
Natural Complex Systems, which are the subject of this book. To fully comprehend how a Complex
System works, it is necessary not only to break it into pieces but also to examine it in its entirety.
Therefore, even the apparently robust epistemological pillar of reductionism sways when we strive
to describe Complex Systems. A Complex System is like an outstanding piece of music or a bril-
liant work of art. If we listen to a Beethoven’s symphony or we look at Raphael’s fresco, we can try
to understand their beauties by decomposing them in their constitutive elements, which are musical
notes and colors, respectively. If the single constituents are taken apart, they are not masterpieces.
Only when they are organized uniquely by the musician in time (symphony) or by the painter in
space (fresco), musical notes and colors become attractive. The same is true for Complex Systems.
Moreover, Complex Systems exhibit as either stationary, periodic, chaotic or stochastic dynamics.
Whatever the kind of dynamics, it is usually extremely sensitive to initial and contour conditions.
This property imposes substantial limitations on the reproducibility of experiments. Many of the
experiments on Complex Systems are historical events that are not reproducible. We can take a
beautiful image from the essay “Of clouds and clocks” written by the philosopher Karl Popper
(1972) and say that science, in the past, had been occupied with clocks, i.e., simple, deterministic
systems having reproducible behaviors. Now, instead, science has to deal with clouds, i.e., Complex
Systems, having unique and hardly replicable behaviors (Figure 1.7).
Complex Systems are everywhere: in physics, chemistry, biology, geology, economy, sociology,
and so on. To try to describe them, we collect a gargantuan amount of data by monitoring accurately
and continuously their dynamics. If we want to handle the Big Data, it is urgent to improve the per-
formances of our electronic computers, contrive more powerful computing machines with brand-
new architectures, and formulate more efficient algorithms. Probably, we need a new methodology
26 An example is a gas consisting of an Avogadro’s number of particles.
27 When we say, “strong interactions” we mean “non-linear interactions.” In fact, any Natural Complex System is characterized by non-linear feedback loops among its constitutive elements. The consequence is that any Natural Complex System is
extremely sensitive to small changes.
Introduction
15
FIGURE 1.7 Simple Systems are epitomized by clocks; Complex Systems by clouds.
of scientific inquiry and a new theory, as well. In other words, we expect a third gateway event that
should usher us in a new scientific era that will likely be called the “Computational Period.”
1.1.4 The “comPuTaTional Period”
The experimental investigation of Complex Systems presents two significant hurdles. First, experi-
ments are often not reproducible. Second, many of the experiments require long periods of time.
Fortunately, there are computational experiments to get around these hurdles. Today, people
working in all areas of Complexity use computers to make experiments, test hypothesis, and gain
insights when no other routes are feasible. “Really efficient high-speed computing devices may, in
the field of nonlinear partial differential equations as well as in many other fields, which are now
difficult or entirely denied access, provide us with those heuristic hints which are needed in all
parts of mathematics for genuine progress” (von Neumann 1966).
A promising strategy to deal with Complex Systems and face the Complexity Challenges
presented in paragraph 1.1 is the interdisciplinary research area of Natural Computing. The funda-
mental idea is that every natural causal phenomenon is a kind of computation because information
is encoded through the states of natural systems. Therefore, every element of the universe works
like a computing machine that processes information. Accordingly, the epistemological pillar of
Mechanism (see Figure 1.6) assumes another aspect: the universe is a vast computational machine
(Lloyd 2006). This new idea brings to the forefront the Information variable. As Murray Gell-Mann
stated, “although Complex Systems differ widely in their physical attributes, they resemble one
another in the way they handle information. That common feature is perhaps the best starting point
for exploring how they operate.” We need a new theory. This new theory should allow us to predict
the emergent properties of Complex Systems. Through it, it should become evident why living
beings have the unique feature of using energy and matter to encode and process information and
exploit this information to make decisions. The new theory will probably include the two funda-
mental attributes of information: the quantity, already rationalized by Claude Shannon (1948), and
the quality, i.e., its semantics. It is not an easy task to rationalize semantic information because it
is context-dependent. During the industrial age, in the eighteenth and nineteenth centuries AD,
entrepreneurs and scientists were engaged in optimizing the performances of thermal machines,
and the laws of “Thermodynamics” have been formulated. Nowadays, scientists, philosophers,
16
Untangling Complex Systems
entrepreneurs, and politicians are all engaged in trying to win the Complexity Challenges, and the
laws of an “Info-dynamics” might be developed. Then, perhaps, Complex Systems would become
more understandable and their investigation less daunting.
1.2 WHAT IS SCIENCE, TODAY?
If we want to contribute to the formulation of new scientific methodologies and theories, it is
fundamental to think how science evolves. If we consider the “Experimental Period,” we may say
that scientific knowledge has grown like a tree. The principal macroscopic components of a tree are
roots, trunk, branches, and leaves. Roots absorb water and nutrients from the soil and feed the rest of
the tree. Leaves harvest two ingredients, light from the Sun and carbon dioxide from the atmosphere,
to synthesize carbohydrates and permit the tree to grow. In the “Tree of Scientific Knowledge”
(see Figure 1.8), the roots are represented by Mathematics. The trunk is made up of Physics and
Chemistry. The branches are the other scientific disciplines, which are grounded in Mathematics,
Physics, and Chemistry. For instance, Geology that deals with the properties and the transforma-
tions of the Earth; Astronomy, whose interests are the properties of the extra-terrestrial objects and
phenomena. Another relevant branch of the “Tree of Scientific Knowledge” is Biology, which is the
study of that fantastic phenomenon that is life on earth. Medicine and Veterinary Science ramify
from Biology, and they focus on the physical health of humans and animals, respectively. Psychology
studies human mind and behavior. Social Sciences, based on Biology and Psychology, concern
about the meta-biological aspects of human behavior. They include Anthropology, Sociology, and
Economics. Anthropology is focused on the definition of human life and its origin. Sociology
studies social structures, their organizations, the rules, and processes binding people and their
institutions. Economics concerns the exploitation of scarce resources and deals with the organiza-
tion of productive activities and interchange of goods to best meet individual and collective needs.
Based mainly on Chemistry, Biology, and Social Sciences, Agricultural science deals with the use
of plants, fungi, and animals for food, fiber, and other products used to sustain life. The application
of physical, chemical, social, and economic knowledge to design and build structures, machines,
devices, systems, and processes is the scope of Engineering.
Social Sciences
Economy
Psychology
Engineering
Veterinary Science
Agriculture
Medicine
Astronomy
Biology
Geology
Chemistry
Physics
Mathematics
Tree of Scientific Knowledge
FIGURE 1.8 Schematic structure of the “Tree of Scientific Knowledge.”
Introduction
17
Legend
Axioms
Inductive Jumps
(1) (6) (11)
(2)
Deductions
(2) (7) (12)
Theorems
(1)
Experiments
(3) (8) (13)
(3)
Technological
(4)
development
(4) (9) (14)
(0) Known Nature
Island
(a)
New axioms
New axioms
(7)
(12)
New theorems
New theorems
(6)
(11)
(13)
(8)
(9)
(14)
(5) Known Nature Island
(10) Known Nature Island
(b)
(c)
FIGURE 1.9 The mechanism of growth of scientific knowledge.
In the “Tree of Scientific Knowledge,” the leaves represent the multitudes of men and women
who have been dedicating their lives to the growth of scientific knowledge. Their nutrients are curi-
osity and the will of improving human welfare, along with the rigorous principles of logic and the
acquired scientific notions.
The mechanism of growth of scientific knowledge is represented schematically in Figure 1.9.
Scientists collect data about natural phenomena by using instruments. The sensitivity and reso-
lution of the available facilities define the boundaries of what can be observed from what remains
unexplored; they outline the extension of the “Known Nature Island” (see graph [a] in Figure 1.9).
From the investigation of the detectable natural phenomena, just a few ingenious scientists can make
inductive jumps (represented by the arrow labeled as [1] in the graph [a] of Figure 1.9) and formulate axioms and postulates. Axioms and postulates28 are the fundamental principles and laws of
scientific knowledge. They are created by induction, i.e., by the free invention of the human mind. 29
28 Both axioms and postulates are assumed to be true without any proof or demonstration. Usually, axioms are evident statements, whereas postulates are not necessarily evident; they are accepted as far as they originate theorems having predicting power.
29 Albert Einstein defined the inductive jumps as “free creation of human mind” in a letter sent to his lifelong friend Maurice Solovine, in 1952. In this letter, Einstein offered a remarkable description of his idea about the scientific methodology. He drew a diagram in which the functions of induction, deduction, and experience were clearly laid out (see the sketch below and von Bayer 2004).
Axioms
Deduced theorems
E
Manifold of immediate experiences
18
Untangling Complex Systems
The first examples of known intuitive figures were Euclid, who formulated the axioms of geom-
etry, and some naturalist philosophers, such as Anaxagoras and Empedocles, who put the principle
of mass conservation forward. Further brilliant intuitive minds were Newton and Leibniz (giv-
ing birth to the Calculus), Clausius (formulating the definition of Entropy and the Second Law of
Thermodynamics to interpret the irreversibility of nature), Darwin (who, after his long trip around
the world on board of the Beagle, formulated the law of Natural Selection as principle that rules
biological evolution), Planck (advancing the idea of Quantization to understand phenomena of the
microscopic world), Schrödinger (formulating the postulates of Quantum Mechanics), Einstein
(developing the Relativistic theory), and the Polish mathematician Benoit Mandelbrot (1924–2010)
(proposing Fractal geometry to describe the shapes of natural objects such as coasts, trees, clouds,
ferns, et cetera).
After the inductive jump (1), the next step in the evolution of scientific knowledge is the deduc-
tion of theorems and propositions from axioms and postulates (see arrow [2] in the graph [a]).
Scientists who deduce theorems are usually named as “theoreticians.” The validity of theorems and
hence, implicitly, of axioms must be proved by designing and performing suitable experiments. The
theoretical predictions must be compared with the results of real experiments (see the arrow labeled
as [3] in the graph [a] of Figure 1.9). Experiments are carried out by those scientists we call “experimentalists.” The main reasoning strategy of experimentalists is abduction consisting of finding
out the causes of phenomena, after detecting the effects and considering the rules. The acquisition
of new scientific knowledge usually promotes technological development. New technologies often
allow deepening the observation of nature. Therefore, science and technology extend the borders of
“Known Nature Island” (see step [4] in the graph [a]). In a larger “Known Nature Island” (see graph
[b] of Figure 1.9), new phenomena and features become evident. Their interpretation often requires the formulation of new axioms through brand-new inductive jumps (step [6] in the graph [b]). The
novel axioms allow the deduction of new theorems and propositions (step [7]). Then, they need to
be experimentally confirmed (step [8]). As seen before, the acquisition of new knowledge induces
further technological development. More powerful technologies extend our observational capabili-
ties. The “Known Nature Island” becomes even more extensive (step [9]) and new surprising details
pop up. For their interpretation, we require a new cycle of intuitions, deductions, and abductions30
accompanied by a permanent dialogue with nature through experiments (graph [c] of Figure 1.9).
And so on, indefinitely.
Now, at the beginning of the twenty-first century, we are scrutinizing the behavior of Complex
Systems, hoping that someone will make a fruitful inductive jump and will formulate those axioms
that we are still waiting. Such an inductive jump is not an easy task because it requires an interdisci-
plinary knowledge. In fact, Natural Complexity is a subject that involves all the disciplines depicted
in Figure 1.8. Such disciplines are now strongly linked as nodes of a network and grow through
mutual connections.
1.3 PURPOSE AND CONTENTS OF THIS BOOK
To favor the awaited inductive jumps in the field of Complexity, we need to prepare the new gen-
erations of students to have interdisciplinary interests and knowledge. This book is born with this
purpose. It has been written by; I dare say, a Philo-physicist with a background in chemistry but
Rule
30 The three types of scientific reasoning about a causal event ( Cause → Effect) are: Induction: Finding the rule when cause and effect are known.
Deduction: Finding the effect when cause and rule are known.
Abduction: Finding the cause when the rule and effect are known
Introduction
19
with multidisciplinary interests. It has been written for graduate and PhD students in chemistry,
who should acquire an interdisciplinary outlook, but also for students in interdisciplinary courses
who want to deepen the role of chemistry in the field of Complexity. To me, the writing of this book
has been like a marvelous journey. I have had the luck of telling the most breathtaking moments of
my trip every year when I teach the subject of Complexity to my students. As it occurs in our daily
life, we undertake an exciting, fruitful, and unforgettable journey whenever one or more queries
guide us. In my case, after noticing that in the outside world there are both self-organizing and cha-
otic phenomena, I have written this book with the intention of finding answers to the two following
Really Big Questions:
1. If the Second Law of Thermodynamics is true, how is it possible to observe the spontane-
ous emergence of order in time and/or space? Is it possible to violate the Second Law?
2. What are the features of the Complex Systems? When and how do the emergent properties
emerge? Can we untangle Complex Systems?
The next three chapters give answers to the first RBQ. Chapter 2 is a thorough analysis of the
Second Law. Chapters 3 and 4 present the theory of non-equilibrium thermodynamics. Then, the theory of non-linear dynamics is introduced by the description of the emergence of temporal order
in ecosystems (Chapter 5), economy (Chapter 6), within a living being (Chapter 7), and in a chemical laboratory (Chapter 8). Chapter 9 describes the emergence of order in space through phenomena such as Turing structures, chemical waves, and periodic precipitations by presenting examples in
chemistry, biology, physics, and geology. Then, Chapter 10 introduces the concept of Chaos in
time, whereas Chapter 11 covers the concept of Chaos in space, by explaining fractals. In Chapters
12 and 13, we are ready to give answers to the second RBQ. Chapter 12 shows the intimate relation between Natural Complexity and Computational Complexity. It outlines the Complexity Challenges
Natural Complex Systems, which are the subject of this book. To fully comprehend how a Complex
System works, it is necessary not only to break it into pieces but also to examine it in its entirety.
Therefore, even the apparently robust epistemological pillar of reductionism sways when we strive
to describe Complex Systems. A Complex System is like an outstanding piece of music or a bril-
liant work of art. If we listen to a Beethoven’s symphony or we look at Raphael’s fresco, we can try
to understand their beauties by decomposing them in their constitutive elements, which are musical
notes and colors, respectively. If the single constituents are taken apart, they are not masterpieces.
Only when they are organized uniquely by the musician in time (symphony) or by the painter in
space (fresco), musical notes and colors become attractive. The same is true for Complex Systems.
Moreover, Complex Systems exhibit as either stationary, periodic, chaotic or stochastic dynamics.
Whatever the kind of dynamics, it is usually extremely sensitive to initial and contour conditions.
This property imposes substantial limitations on the reproducibility of experiments. Many of the
experiments on Complex Systems are historical events that are not reproducible. We can take a
beautiful image from the essay “Of clouds and clocks” written by the philosopher Karl Popper
(1972) and say that science, in the past, had been occupied with clocks, i.e., simple, deterministic
systems having reproducible behaviors. Now, instead, science has to deal with clouds, i.e., Complex
Systems, having unique and hardly replicable behaviors (Figure 1.7).
Complex Systems are everywhere: in physics, chemistry, biology, geology, economy, sociology,
and so on. To try to describe them, we collect a gargantuan amount of data by monitoring accurately
and continuously their dynamics. If we want to handle the Big Data, it is urgent to improve the per-
formances of our electronic computers, contrive more powerful computing machines with brand-
new architectures, and formulate more efficient algorithms. Probably, we need a new methodology
26 An example is a gas consisting of an Avogadro’s number of particles.
27 When we say, “strong interactions” we mean “non-linear interactions.” In fact, any Natural Complex System is characterized by non-linear feedback loops among its constitutive elements. The consequence is that any Natural Complex System is
extremely sensitive to small changes.
Introduction
15
FIGURE 1.7 Simple Systems are epitomized by clocks; Complex Systems by clouds.
of scientific inquiry and a new theory, as well. In other words, we expect a third gateway event that
should usher us in a new scientific era that will likely be called the “Computational Period.”
1.1.4 The “comPuTaTional Period”
The experimental investigation of Complex Systems presents two significant hurdles. First, experi-
ments are often not reproducible. Second, many of the experiments require long periods of time.
Fortunately, there are computational experiments to get around these hurdles. Today, people
working in all areas of Complexity use computers to make experiments, test hypothesis, and gain
insights when no other routes are feasible. “Really efficient high-speed computing devices may, in
the field of nonlinear partial differential equations as well as in many other fields, which are now
difficult or entirely denied access, provide us with those heuristic hints which are needed in all
parts of mathematics for genuine progress” (von Neumann 1966).
A promising strategy to deal with Complex Systems and face the Complexity Challenges
presented in paragraph 1.1 is the interdisciplinary research area of Natural Computing. The funda-
mental idea is that every natural causal phenomenon is a kind of computation because information
is encoded through the states of natural systems. Therefore, every element of the universe works
like a computing machine that processes information. Accordingly, the epistemological pillar of
Mechanism (see Figure 1.6) assumes another aspect: the universe is a vast computational machine
(Lloyd 2006). This new idea brings to the forefront the Information variable. As Murray Gell-Mann
stated, “although Complex Systems differ widely in their physical attributes, they resemble one
another in the way they handle information. That common feature is perhaps the best starting point
for exploring how they operate.” We need a new theory. This new theory should allow us to predict
the emergent properties of Complex Systems. Through it, it should become evident why living
beings have the unique feature of using energy and matter to encode and process information and
exploit this information to make decisions. The new theory will probably include the two funda-
mental attributes of information: the quantity, already rationalized by Claude Shannon (1948), and
the quality, i.e., its semantics. It is not an easy task to rationalize semantic information because it
is context-dependent. During the industrial age, in the eighteenth and nineteenth centuries AD,
entrepreneurs and scientists were engaged in optimizing the performances of thermal machines,
and the laws of “Thermodynamics” have been formulated. Nowadays, scientists, philosophers,
16
Untangling Complex Systems
entrepreneurs, and politicians are all engaged in trying to win the Complexity Challenges, and the
laws of an “Info-dynamics” might be developed. Then, perhaps, Complex Systems would become
more understandable and their investigation less daunting.
1.2 WHAT IS SCIENCE, TODAY?
If we want to contribute to the formulation of new scientific methodologies and theories, it is
fundamental to think how science evolves. If we consider the “Experimental Period,” we may say
that scientific knowledge has grown like a tree. The principal macroscopic components of a tree are
roots, trunk, branches, and leaves. Roots absorb water and nutrients from the soil and feed the rest of
the tree. Leaves harvest two ingredients, light from the Sun and carbon dioxide from the atmosphere,
to synthesize carbohydrates and permit the tree to grow. In the “Tree of Scientific Knowledge”
(see Figure 1.8), the roots are represented by Mathematics. The trunk is made up of Physics and
Chemistry. The branches are the other scientific disciplines, which are grounded in Mathematics,
Physics, and Chemistry. For instance, Geology that deals with the properties and the transforma-
tions of the Earth; Astronomy, whose interests are the properties of the extra-terrestrial objects and
phenomena. Another relevant branch of the “Tree of Scientific Knowledge” is Biology, which is the
study of that fantastic phenomenon that is life on earth. Medicine and Veterinary Science ramify
from Biology, and they focus on the physical health of humans and animals, respectively. Psychology
studies human mind and behavior. Social Sciences, based on Biology and Psychology, concern
about the meta-biological aspects of human behavior. They include Anthropology, Sociology, and
Economics. Anthropology is focused on the definition of human life and its origin. Sociology
studies social structures, their organizations, the rules, and processes binding people and their
institutions. Economics concerns the exploitation of scarce resources and deals with the organiza-
tion of productive activities and interchange of goods to best meet individual and collective needs.
Based mainly on Chemistry, Biology, and Social Sciences, Agricultural science deals with the use
of plants, fungi, and animals for food, fiber, and other products used to sustain life. The application
of physical, chemical, social, and economic knowledge to design and build structures, machines,
devices, systems, and processes is the scope of Engineering.
Social Sciences
Economy
Psychology
Engineering
Veterinary Science
Agriculture
Medicine
Astronomy
Biology
Geology
Chemistry
Physics
Mathematics
Tree of Scientific Knowledge
FIGURE 1.8 Schematic structure of the “Tree of Scientific Knowledge.”
Introduction
17
Legend
Axioms
Inductive Jumps
(1) (6) (11)
(2)
Deductions
(2) (7) (12)
Theorems
(1)
Experiments
(3) (8) (13)
(3)
Technological
(4)
development
(4) (9) (14)
(0) Known Nature
Island
(a)
New axioms
New axioms
(7)
(12)
New theorems
New theorems
(6)
(11)
(13)
(8)
(9)
(14)
(5) Known Nature Island
(10) Known Nature Island
(b)
(c)
FIGURE 1.9 The mechanism of growth of scientific knowledge.
In the “Tree of Scientific Knowledge,” the leaves represent the multitudes of men and women
who have been dedicating their lives to the growth of scientific knowledge. Their nutrients are curi-
osity and the will of improving human welfare, along with the rigorous principles of logic and the
acquired scientific notions.
The mechanism of growth of scientific knowledge is represented schematically in Figure 1.9.
Scientists collect data about natural phenomena by using instruments. The sensitivity and reso-
lution of the available facilities define the boundaries of what can be observed from what remains
unexplored; they outline the extension of the “Known Nature Island” (see graph [a] in Figure 1.9).
From the investigation of the detectable natural phenomena, just a few ingenious scientists can make
inductive jumps (represented by the arrow labeled as [1] in the graph [a] of Figure 1.9) and formulate axioms and postulates. Axioms and postulates28 are the fundamental principles and laws of
scientific knowledge. They are created by induction, i.e., by the free invention of the human mind. 29
28 Both axioms and postulates are assumed to be true without any proof or demonstration. Usually, axioms are evident statements, whereas postulates are not necessarily evident; they are accepted as far as they originate theorems having predicting power.
29 Albert Einstein defined the inductive jumps as “free creation of human mind” in a letter sent to his lifelong friend Maurice Solovine, in 1952. In this letter, Einstein offered a remarkable description of his idea about the scientific methodology. He drew a diagram in which the functions of induction, deduction, and experience were clearly laid out (see the sketch below and von Bayer 2004).
Axioms
Deduced theorems
E
Manifold of immediate experiences
18
Untangling Complex Systems
The first examples of known intuitive figures were Euclid, who formulated the axioms of geom-
etry, and some naturalist philosophers, such as Anaxagoras and Empedocles, who put the principle
of mass conservation forward. Further brilliant intuitive minds were Newton and Leibniz (giv-
ing birth to the Calculus), Clausius (formulating the definition of Entropy and the Second Law of
Thermodynamics to interpret the irreversibility of nature), Darwin (who, after his long trip around
the world on board of the Beagle, formulated the law of Natural Selection as principle that rules
biological evolution), Planck (advancing the idea of Quantization to understand phenomena of the
microscopic world), Schrödinger (formulating the postulates of Quantum Mechanics), Einstein
(developing the Relativistic theory), and the Polish mathematician Benoit Mandelbrot (1924–2010)
(proposing Fractal geometry to describe the shapes of natural objects such as coasts, trees, clouds,
ferns, et cetera).
After the inductive jump (1), the next step in the evolution of scientific knowledge is the deduc-
tion of theorems and propositions from axioms and postulates (see arrow [2] in the graph [a]).
Scientists who deduce theorems are usually named as “theoreticians.” The validity of theorems and
hence, implicitly, of axioms must be proved by designing and performing suitable experiments. The
theoretical predictions must be compared with the results of real experiments (see the arrow labeled
as [3] in the graph [a] of Figure 1.9). Experiments are carried out by those scientists we call “experimentalists.” The main reasoning strategy of experimentalists is abduction consisting of finding
out the causes of phenomena, after detecting the effects and considering the rules. The acquisition
of new scientific knowledge usually promotes technological development. New technologies often
allow deepening the observation of nature. Therefore, science and technology extend the borders of
“Known Nature Island” (see step [4] in the graph [a]). In a larger “Known Nature Island” (see graph
[b] of Figure 1.9), new phenomena and features become evident. Their interpretation often requires the formulation of new axioms through brand-new inductive jumps (step [6] in the graph [b]). The
novel axioms allow the deduction of new theorems and propositions (step [7]). Then, they need to
be experimentally confirmed (step [8]). As seen before, the acquisition of new knowledge induces
further technological development. More powerful technologies extend our observational capabili-
ties. The “Known Nature Island” becomes even more extensive (step [9]) and new surprising details
pop up. For their interpretation, we require a new cycle of intuitions, deductions, and abductions30
accompanied by a permanent dialogue with nature through experiments (graph [c] of Figure 1.9).
And so on, indefinitely.
Now, at the beginning of the twenty-first century, we are scrutinizing the behavior of Complex
Systems, hoping that someone will make a fruitful inductive jump and will formulate those axioms
that we are still waiting. Such an inductive jump is not an easy task because it requires an interdisci-
plinary knowledge. In fact, Natural Complexity is a subject that involves all the disciplines depicted
in Figure 1.8. Such disciplines are now strongly linked as nodes of a network and grow through
mutual connections.
1.3 PURPOSE AND CONTENTS OF THIS BOOK
To favor the awaited inductive jumps in the field of Complexity, we need to prepare the new gen-
erations of students to have interdisciplinary interests and knowledge. This book is born with this
purpose. It has been written by; I dare say, a Philo-physicist with a background in chemistry but
Rule
30 The three types of scientific reasoning about a causal event ( Cause → Effect) are: Induction: Finding the rule when cause and effect are known.
Deduction: Finding the effect when cause and rule are known.
Abduction: Finding the cause when the rule and effect are known
Introduction
19
with multidisciplinary interests. It has been written for graduate and PhD students in chemistry,
who should acquire an interdisciplinary outlook, but also for students in interdisciplinary courses
who want to deepen the role of chemistry in the field of Complexity. To me, the writing of this book
has been like a marvelous journey. I have had the luck of telling the most breathtaking moments of
my trip every year when I teach the subject of Complexity to my students. As it occurs in our daily
life, we undertake an exciting, fruitful, and unforgettable journey whenever one or more queries
guide us. In my case, after noticing that in the outside world there are both self-organizing and cha-
otic phenomena, I have written this book with the intention of finding answers to the two following
Really Big Questions:
1. If the Second Law of Thermodynamics is true, how is it possible to observe the spontane-
ous emergence of order in time and/or space? Is it possible to violate the Second Law?
2. What are the features of the Complex Systems? When and how do the emergent properties
emerge? Can we untangle Complex Systems?
The next three chapters give answers to the first RBQ. Chapter 2 is a thorough analysis of the
Second Law. Chapters 3 and 4 present the theory of non-equilibrium thermodynamics. Then, the theory of non-linear dynamics is introduced by the description of the emergence of temporal order
in ecosystems (Chapter 5), economy (Chapter 6), within a living being (Chapter 7), and in a chemical laboratory (Chapter 8). Chapter 9 describes the emergence of order in space through phenomena such as Turing structures, chemical waves, and periodic precipitations by presenting examples in
chemistry, biology, physics, and geology. Then, Chapter 10 introduces the concept of Chaos in
time, whereas Chapter 11 covers the concept of Chaos in space, by explaining fractals. In Chapters
12 and 13, we are ready to give answers to the second RBQ. Chapter 12 shows the intimate relation between Natural Complexity and Computational Complexity. It outlines the Complexity Challenges
