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**These are excerpts and elaborations from my book "The Nature of Consciousness"**

Both biological and physical
sciences need a mathematical model of phenomena of emergence (spontaneous
creation of order), and in particular adaptation, as well as a physical
justification of their dynamics (which seems to violate physical laws such as
the second law of Thermodynamics). The French physicist Sadi
Carnot, one of the founding fathers of Thermodynamics, realized that the
statistical behavior of a complex system could be predicted if its parts were
all identical and their interactions weak.
It was not feasible for Physics to predict the behavior of each single
part, but it was relatively easy to predict the statistical behavior of the
parts. At the beginning of the 20 The most interesting feature
of chaotic systems is that under some circumstances chaotic systems
spontaneously "crystallize" into a higher degree of order, i.e.
properties begin to emerge. A system
must be “complex” enough for any property to “emerge” out of it. Complexity can be formally
defined as nonlinearity. Nonlinearity is best visualized as a game in which
playing the game changes the rules. In a linear system an action has an effect
that is somewhat proportionate to the effect. In a nonlinear system an action
has an effect that causes a change in the action itself. As a matter of fact,
the world is mostly nonlinear. The “linear” world studied by classical Physics
(Newton’s equations are linear) is really just an exception to the rule. The science of nonlinear
dynamics is also known as "chaos theory", from the definition
introduced by US physicist James Yorke ("Period Three Implies
Chaos", 1975), because unpredictable solutions emerge from nonlinear
equations; in other words, they appear to behave in random fashion. The US mathematician Edward
Lorenz (“Deterministic Nonperiodic
Flow”, 1963) was the first one to focus on the equations about deterministic
laws that generated unpredictable behavior. These equations were unpredictable
because a slight change in the initial conditions had devastating consequences.
In linear equations the effects are proportional to the causes. Linear
equations are modular in the sense that one can reduce them to simpler
equations and then reassemble the solution. Unfortunately, linearity is an
ideal state that usually happens only when a system is close to equilibrium.
Real-world systems are rarely close to equilibrium. Anything that is alive, in
particular, and anything that is evolving is in a state of non-equilibrium.
Nonlinear equations cannot be reduced to simpler equations: the whole is
literally more than its parts. This “synergistic” aspect is a sort of internal
loop, and is responsible for the property that a small change in the initial
conditions can grow exponentially quickly, the so called “butterfly effect”
named after an Edward Lorenz lecture ("Does The Flap Of A Butterfly's Wings
In Brazil Set Off A Tornado In Texas?”, 1972). Nonlinearity tends to show
up when one tries to explain how global behavior emerges from local behavior.
For example, it's easy to muster the rules of dynamics and electromagnetism
when applied to isolated systems but difficult to predict the weather of a
region. It's easy to muster the arithmetic and statistical formulas of economic
principles but not to predict what will happen to a nation's economy. The
reason is that these complex systems have components that influence each other.
Chaos is not intractable
though. As Stephen Smale showed ("Differentiable dynamical systems", 1967), an
irregularity can persist in a chaotic system to the point that the system is
"stable". A useful abstraction to
describe the evolution of a system in time is that of a "phase
space". Our ordinary space has only three dimensions (width, height,
depth) but in theory we can think of spaces with any number of dimensions. A
phase space has six dimensions, three of which are the usual spatial dimensions
while the other three are the components of velocity along those spatial
dimensions. In ordinary 3-dimensional space, a "point" can only
represent the position of a system. In 6-dimensional phase space, a point
represents both the position and the motion of the system. The evolution of a
system is represented by some sort of shape in phase space. An “attractor” is a region
of phase space where the system is doomed to end up eventually. For a linear
system the attractor basically describes its typical behavior. A point
attractor is the state in which the system ends its motion. A periodic
attractor is the state that a system returns to periodically. Chaotic systems
do not have point or periodic attractors, but chaotic systems may exhibit
attractors that are shapes with fractional dimension. If the region that these
“strange attractors” occupy is finite, then, by definition, the behavior of the
chaotic system is not truly random. The first “strange attractor” was “discovered”
by the US meteorologist Edward Lorenz (“Deterministic Nonperiodic
Flow”, 1963): it was due to a nonlinear system that evolves over time in a
non-repeating pattern. In phase space, its evolution looks like an infinite
series of ever-changing patterns that seem to be attracted to a point but never
repeat the same trajectory around it. The US physicist Robert Shaw ("Strange Attractors,
Chaotic Behavior, and Information Flow", 1981) realized that strange
attractors increase entropy, i.e. "create" information. They create
information about the beahvior of a chaotic system where no information about it was available. Nonlinear processes are
ubiquitous. They are processes of emergent order and complexity, of how structure
arises from the interaction of many independent units. These processes recur at
every level, from morphology to behavior. At every level of science (including
the brain and life) the spontaneous emergence of order, or self-organization of
complex systems, is a common theme. One feature of chaos,
discovered by the Australian mathematician Robert May (""Simple Mathematics
Models with very complicated dynamics", 1976), is “self-similarity”, the
fact that the “chaotic” behavior sometimes embeds an exact replica of itself.
By studying self-similarity, the Polish-born mathematician Benoit Mandelbrot came up with the idea of
fractal geometry (“Fractal Objects”, 1975). Euclidean geometry fails to capture
the essence of ordinary natural shapes
that are way more complex than straight lines or perfect circles. Mandelbrot introduced fractional
dimensions to express the degree of “irregularity” of a shape (for example, of
a coastline). The US physicist Mitchell
Feigenbaum discovered a new universal
constant that applies to chaotic systems by analyzing how a nonlinear system
becomes chaotic ("Quantitative Universality for a Class of Nonlinear
Transformations", 1978). At some point the behavior of the system splits
into two (a “bifurcation”) and then into four and so forth, at an increasingly
faster rate. It is this acceleration of bifurcations that defines “chaos”.
Feigenbaum discovered that the ratio between the bifurcations is 4.669. There is order even in chaos. Darwin's vision of natural selection as a creator of order is probably not
sufficient to explain all the spontaneous order exhibited by both living and
inanimate matter. There might be other physical principles at work. Koestler and Salthe showed how complexity entails
hierarchical organization. Von Bertalanffy's general systems theory, Haken's synergetics, and Prigogine's non-equilibrium Thermodynamics belong to the class of theories that
extend Physics to dynamic systems. These theories have in
common the fact that they deal with self-organization (how collections of parts
can produce structures) and try to provide a unified view of the universe at
different levels of organization (from living organisms to physical systems to
societies). The drawback in any study of
complex systems is that there is no commonly accepted definition of complexity
and method of measuring complexity. Kolmogorov’s complexity is not a biologist’s complexity. A biologist does not
know how to compare the complexity of a tomato with the complexity of the stock
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