Chaos Theorie (in Englisch)
Schlagwörter:
Chaos Theorie (in Englisch) Mathematik, Referat, Hausaufgabe, Chaos Theorie (in Englisch)
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Chaos Theorie (in Englisch) Mathematik, Referat, Hausaufgabe, Chaos Theorie (in Englisch)
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Referat
Chaos Theory
Dialectical materialism, elaborated by Karl Marx and
Frederick Engels, was concerned with much more than political economy: it was a
world view. Nature, as Engels in particular sought to demonstrate in his
writings, is proof of the correctness of both materialism and dialectics. "My
recapitulation of mathematics and the natural sciences," he wrote, "was
undertaken in order to convince myself also in detail...that in nature amid the
welter of innumerable changes, the same dialectical laws of motion force their
way through as those which in history govern the apparent fortuitousness of
events..." (16)
Since their day, every important new advance in
scientific discovery has confirmed the Marxian outlook although scientists,
because of the political implications of an association with Marxism, seldom
acknowledge dialectical materialism. Now, the advent of chaos theory provides
fresh backing for the fundamental ideas of the founders of scientific socialism.
Up to now chaos has been largely ignored by scientists, except as a nuisance or
something to be avoided. A tap drips, sometimes regularly, sometimes not; the
movement of a fluid is either turbulent or not; the heart beats regularly but
sometimes goes into a fibrillation; the weather blows hot or cold. Wherever
there is motion that appears to be chaotic—and it is all around
us—there is generally little attempt to come to terms with it from a
strictly scientific point of view.
What then, are the general features of chaotic systems?
Having described them in mathematical terms, what application does the
mathematics have? One of the features given prominence by Gleick and others is
what has been dubbed "the butterfly effect." Lorenz, had discovered on his
computer-simulated weather a remarkable development. One of his simulations was
based on twelve variables, including, as we said, non-linear relationships. He
found that if he started his simulation with values that were only slightly
different from the original—the difference being that one set were down to
six decimal places and the second set down three places—then the "weather"
produced by the computer soon veered wildly from the original. Where perhaps a
slight perturbation might have been expected, there was, only after a brief
period of recognisable similarity, a completely different
pattern.
This means that in a complex, non-linear system, a small
change in the input could produce a huge change in the output. In Lorenz’s
computer world, it was equivalent to a butterfly’s wing-beat causing a
hurricane in another part of the world; hence the expression. The conclusion
that can be drawn from this is that, given the complexity of the forces and
processes that go to determine the weather, it can never be predicted beyond a
short period of time ahead. In fact, the biggest weather computer in the world,
in the European centre for Medium-range Weather Forecasting, does as many as 400
million calculations every second. It is fed 100 million separate weather
measurements from around the world every day, and it processes data in three
hours of continuous running, to produce a ten day forecast. Yet beyond two or
three days the forecasts are speculative, and beyond six or seven they are
worthless. Chaos theory, then, sets definite limits to the predictability of
complex non-linear systems.
It is strange, nevertheless, that Gleick and others have
paid so much attention to the butterfly effect, as if it injects a strange
mystique into chaos theory. It is surely well established (if not accurately
modelled mathematically) that in other similarly complex systems a small input
can produce a large output, that an accumulation of "quantity" can be
transformed to "quality." There is only a difference of less than two per cent,
for example, in the basic genetic make-up of human beings and
chimpanzees—a difference that can be quantified in terms of molecular
chemistry. Yet in the complex, non-linear processes that are involved in
translating the genetic "code" into a living animal, this small dissimilarity
means the difference between one species and another.
Marxism applies itself to perhaps the most complex of
all non-linear systems—human society. With the colossal interaction of
countless individuals, politics and economics constitute so complex a system
that alongside it, the planet’s weather systems looks like clockwork.
Nevertheless, as is the case with other "chaotic" systems, society can be
treated scientifically—as long as the limits, like the weather, are
understood. Unfortunately, Gleick’s book is not clear on the application
of chaos theory to politics and economics. He cites an exercise by Mandelbrot,
who fed his IBM computer with a hundred year’s worth of cotton prices from
the New York exchange. "Each particular price change was random and
unpredictable," he writes. "But the sequence of changes was independent of
scale: curves for daily and monthly price changes matched...the degree of
variation had remained constant over a tumultuous 60-year period that saw two
world wars and a depression." (17)
This passage cannot be taken on face value. It may be
true that within certain limits, it is possible to see the same mathematical
patterns that have been identified in other models or chaotic systems. But given
the almost limitless complexity of human society and economics, it is
inconceivable that major events like wars would not disrupt these patterns.
Marxists would argue that society does lend itself to scientific study. In
contrast to those who see only formlessness, Marxists see human development from
the starting point of material forces, and a scientific description of social
categories like classes, and so on. If the development of chaos science leads to
an acceptance that the scientific method is valid in politics and economics,
then it is a valuable plus. However, as Marx and Engels have always understood,
theirs is an inexact science, meaning that broad trends and developments could
be traced, but detailed and intimate knowledge of all influences and conditions
is not possible.
Cotton prices notwithstanding, the book gives no
evidence that this Marxist view is wrong. In fact, there is no explanation as to
why Mandelbrot apparently saw a pattern in only 60 years’ prices when he
had over 100 years’ of data to play with. In addition, elsewhere in the
book, Gleick adds that "economists have looked for strange attractors in stock
market trends but so far had not found them." Despite the apparent limitations
in the fields of economics and politics, however, it is clear that the
mathematical "taming" of what were thought to be random or chaotic systems has
profound implications for science as a whole. It opens up many vistas for the
study of processes that were largely out of bounds in the past.
Διϖισιον οφ Λαβουρ
One of the main characteristics of the great scientists
of the Renaissance was that they were whole human beings. They had an
all-rounded development, which enabled, for example, Leonardo da Vinci to be a
great engineer, mathematician and mechanician, as well as an artist of genius.
The same was true of Dührer, Machiavelli, Luther, and countless others, of
whom Engels wrote:
"The heroes of that time were not yet in thrall to the
division of labour, the restricting effects of which, with its production of
one-sidedness, we so often notice in their successors." (18) The division of
labour, of course, plays a necessary role in the development of the productive
forces. However, under capitalism, this has been carried to such an extreme that
it begins to turn into its opposite.
The extreme division, on the one hand, between mental
and manual labour means that millions of men and women are reduced to a life of
unthinking drudgery on the production line, denied of any possibility to display
the creativity and inventiveness which is latent in every human being. At the
other extreme, we have the development of a kind of intellectual priestly caste
which has arrogated to itself the sole right to the title of "guardians of
science and culture." To the degree that these people become remote from the
real life of society, this has a negative effect on their consciousness. They
develop in an entirely narrow, one-sided way. Not only is there an abyss
separating "artists" from scientists, but the scientific community itself is
riven with ever-increasing divisions between increasingly narrow
specialisations. It is ironic that, precisely when the "lines of demarcation"
between physics, chemistry and biology are breaking down, the gulf which divides
even different branches of, say, physics has become virtually
unbridgeable.
James Gleick describes the situation
thus:
"Few laymen realise how tightly compartmentalised the
scientific community had become, a battleship with bulkheads sealed against
leaks. Biologists had enough to read without keeping up with the mathematical
literature—for that matter, molecular biologists had enough to read
without keeping up with population biology, physicists had better ways to spend
their time than sifting through the meteorology journals."
In recent years, the advent of chaos theory is one of
the indications that something is beginning to change in the scientific
community. Increasingly, scientists from different fields feel that they have
somehow reached a dead end. It is necessary to break out in a new direction. The
birth of chaos mathematics, therefore, is a proof as Engels would have said, of
the dialectical character of nature, a reminder that reality consists of whole
dynamic systems, or even one whole system, and not of models (however useful)
abstracted from them. What are the main features of chaos theory? Gleick
describes them in the following way:
"To some physicists, chaos is a science of process
rather than state, of becoming rather than being."
"They feel that they are turning back a trend in science
towards reductionism, the analysis of systems in terms of their constituent
parts: quarks, chromosomes, or neutrons. They believe that they are looking for
the whole."
The method of dialectical materialism is precisely to
look at "process rather than state, of becoming rather than being." "More and
more over the past decade, he’d begun to sense that the old reductionist
approaches were reaching a dead end, and that even some of the hard-core
physical scientists were getting fed up with mathematical abstractions that
ignored the real complexities of the world. They seemed to be half-consciously
groping for a new approach—and in the process, he thought, they were
cutting across the traditional boundaries in a way they hadn’t done in
years. Maybe centuries." (19)
Because chaos is a science of whole dynamic systems,
rather than separate parts, it represents, in effect, an unacknowledged
vindication of the dialectical view. Up to now, scientific investigation has
been too much isolated into its constituent parts. In pursuit of the "parts" the
scientific specialist becomes too specialised not infrequently losing all sight
of the "whole." Experimentation and theoretical rationalisations thus became
increasingly removed from reality. More than a century ago, Engels criticised
the narrowness of what he called the metaphysical method, which consisted of
looking at things in an isolated way, which lost sight of the whole. The
starting point of the supporters of chaos theory was a reaction against
precisely this method, which they call "reductionism." Engels explained that the
"reduction" of the study of nature to separate disciplines is to some extent
necessary and inevitable.
"When we reflect on nature or the history of mankind or
our own intellectual activity, at first we see the picture of an endless maze of
connections in which nothing remains what, where and as it was, but everything
moves, changes, comes into being and passes away...
"But this conception, correctly as it expresses the
general character of the picture of phenomena as a whole, does not suffice to
explain the details of which this picture is made up, and so long as we cannot
do this, we are not clear about the whole picture. In order to understand these
details we must detach them from their natural or historical connection and
examine each one separately according to its nature, special causes and effects,
etc."
But as Engels warned, too great a retreat into
"reductionism" can lead to an undialectical view, or a drift to metaphysical
ideas.
"The analysis of nature into its individual parts, the
division of the different natural processes and objects into definite classes,
the study of the internal anatomy of organic bodies in their manifold
forms—these were the fundamental conditions for the gigantic strides in
our knowledge of nature that have been made during the last four hundred years.
But this has bequeathed us the habit of observing natural objects and processes
in isolation, detached from the general context; of observing them not in their
motion, but in their state of rest; not as essentially variable elements, but as
constant ones; not in their life, but in their death." (20)
Now compare this with the following passage from
Gleick’s book:
"Scientists break things apart and look at them one at a
time. If they want to examine the interaction of subatomic particles, they put
two or three together. There is complication enough. The power of
self-similarity, though, begins at much greater levels of complexity. It is a
matter of looking at the whole." (21)
If we substitute the word "reductionism" for "the
metaphysical mode of thought," we see that the central idea is identical. Now
see what conclusion Engels drew from his criticism of reductionism ("the
metaphysical method"):
"But for dialectics, which grasps things and their
images, ideas, essentially in their interconnection, in their sequence, their
movement, their birth and death, such processes as those mentioned above are so
many corroborations of its own method of treatment. Nature is the test of
dialectics, and it must be said for modern natural science that it has furnished
extremely rich and daily increasing materials for this test, and has thus proved
that in the last analysis Nature’s process is dialectical and not
metaphysical.
"But the scientists who have learnt to think
dialectically are still few and far between, and hence the conflict between the
discoveries made and the old traditional mode of thought is the explanation of
the boundless confusion which now reigns in theoretical natural science and
reduces both teachers and students, writers and readers to despair."
(22)
Over one hundred years ago, old Engels accurately
describes the state of the physical sciences today. This is acknowledged by Ilya
Prigogine (Nobel-prize winner for chemistry 1977) and Isabelle Stengers in their
book Order Out of Chaos, Man’s New Dialogue with Nature, where they writes
the following:
"To a certain extent, there is an analogy between this
conflict (between Newtonian physics and the new scientific ideas) and the one
that gave rise to dialectical materialism...The idea of a history of nature as
an integral part of materialism was asserted by Marx and, in greater detail, by
Engels. Contemporary developments in physics, the discovery of the constructive
role played by irreversibility, have thus raised within the natural sciences a
question that has long been asked by materialists. For them, understanding
nature meant understanding it as being capable of producing man and his
societies.
"Moreover, at the time Engels wrote his Dialectics of
Nature, the physical sciences seemed to have rejected the mechanistic world view
and drawn closer to the idea of an historical development of nature. Engels
mentions three fundamental discoveries: energy and the laws governing its
qualitative transformations, the cell as the basic constituent of life, and
Darwin’s discovery of the evolution of species. In view of these great
discoveries, Engels came to the conclusion that the mechanistic world view was
dead." (23)
Despite all the wonderful advances of science and
technology, there is a deep-seated feeling of malaise. An increasing number of
scientists are beginning to rebel against the prevailing orthodoxies and seek
new solutions to the problems facing them. Sooner or later, this is bound to
result in a new revolution in science, similar to the one effected by Einstein
and Planck nearly a century ago. Significantly, Einstein himself was far from
being a member of the scientific establishment.
"The mainstream for most of the twentieth century,"
Gleick remarks, "has been particle physics, exploring the building blocks of
matter at higher and higher energies, smaller and smaller scale, shorter and
shorter times. Out of particle physics have come theories about the fundamental
forces of nature and about the origin of the universe. Yet some young physicists
have grown dissatisfied with the direction of the most prestigious of sciences.
Progress has begun to seem slow, the naming of new particles futile, the body of
theory cluttered. With the coming of chaos, younger scientists believed they
were seeing the beginnings of a course change for all of physics. The field had
been dominated long enough, they felt, by the glittering abstractions of
high-energy particles and quantum mechanics."
Χηαοσ ανδ Διαλεχτιχσ
It is as yet too early to form a definitive view of
chaos theory. However, what is clear is that these scientists are groping in the
direction of a dialectical view of nature. For example, the dialectical law of
the transformation of quantity into quality (and vice versa) plays a prominent
sole in chaos theory:
"He (Von Neumann) recognised that a complicated
dynamical system could have points of instability—critical points where a
small push can have large consequences, as with a ball balanced at the top of a
hill."
And again:
"In science as in life, it is well known that a chain of
events can have a point of crisis that could magnify small changes. But chaos
meant that such points were everywhere. They were pervasive."
(24)
These and many other passages reveal a striking
resemblance between certain aspects of chaos theory and dialectics. Yet the most
incredible thing is that most of the pioneers of "chaos" seem to have not the
slightest knowledge not only of the writings of Marx and Engels, but even of
Hegel! In one sense, this provides even more striking confirmation of the
correctness of dialectical materialism. But in another, it is a frustrating
thought that the absence of an adequate philosophical framework and methodology
has been denied to science needlessly and for such a long time.
For 300 years, physics was based on linear systems. The
name linear refers to the fact that if you plot such an equation on a graph, it
emerges as a straight line. Indeed, much of nature appears to work precisely in
this way. This is why classical mechanics is able to describe it adequately.
However, much of nature is not linear, and cannot be understood through linear
systems. The brain certainly does not function in a linear manner, nor does the
economy, with its chaotic cycle of booms and slumps. A non-linear equation is
not expressed in a straight line, but takes into account the irregular,
contradictory and frequently chaotic nature of reality.
"All this makes me feel very unhappy about cosmologists
who tell us that they’ve got the origins of the Universe pretty well
wrapped up, except for the first millisecond or so of the Big Bang. And with
politicians who assure us that not only is a solid dose of monetarism going to
be good for us, but they’re so certain about it that a few million
unemployed must be just a minor hiccup. The mathematical ecologist Robert May
voiced similar sentiments in 1976. ‘Not only in research, but in the
everyday world of politics and economics, we would all be better off if more
people realised that simple systems do not necessarily possess simple dynamical
properties.’" (25)
The problems of modern science could be overcome far
more easily by adopting a conscious (as opposed to an unconscious, haphazard,
empirical) dialectical method. It is clear that the general philosophical
implications of chaos theory are disputed by its scientists. Gleick quotes Ford,
"a self-proclaimed evangelist of chaos" as saying that chaos means "systems
liberated to randomly explore their every dynamic possibility..." Others refer
to apparently random systems. Perhaps the best definition comes from Jensen, a
theoretical physicist at Yale, who defines "chaos" as "the irregular,
unpredictable behaviour of deterministic, non-linear dynamical
systems."
Rather than elevate randomness to a principle of nature,
as Ford seems to do, the new science does the opposite: it shows irrefutably
that processes that were considered to be random (and may still be so
considered, for everyday purposes) are nevertheless driven by an underlying
determinism—not the crude mechanical determinism of the 18th century but
dialectical determinism.
Some of the claims being made for the new science are
very grand, and with the refinement and development of methods and techniques,
may well prove true. Some of its exponents go so far as to say that the 20th
century will be known for three things: relativity, quantum mechanics and chaos.
Albert Einstein, although one of the founders of quantum theory, was never
reconciled to the idea of a non-deterministic universe. In a letter to the
physicist Neils Bohr, he insisted that "God does not play dice." Chaos theory
has not only shown Einstein to be correct on this point, but even in its
infancy, it is a brilliant confirmation of the fundamental world view put
forward by Marx and Engels over a hundred years ago.
It is really astonishing that so many of the advocates
of chaos theory, who are attempting to break with the stultifying "linear"
methodology and work out a new "non-linear" mathematics, which is more in
consonance with the turbulent reality of ever-changing nature, appear to be
completely unaware of the only genuine revolution in logic in two
millennia—the dialectical logic elaborated by Hegel, and subsequently
perfected on a scientific and materialist basis by Marx and Engels. How many
errors, blind alleys and crises in science could have been avoided if scientists
had been equipped with a methodology which genuinely reflects the dynamic
reality of nature, instead of conflicting with it at every
turn!
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