A talk for the National Collegiate Honors Council,
San Francisco, 31 October, 1996
by
Ralph H. Abraham, abraham@vismath.org
Visual Math Institute, http://www.vismath.org
Abstract. We support the idea that we are now at
a major hinge of history. We review the basic
concepts of the mathematical theories of chaos and
bifurcations in complex dynamical systems,
and apply them to our present moment in history,
the chaos revolution, and our future in the chaos
millennium.
Dedicated to the General Evolution Research Group
and the Lindisfarne Association
1. Introduction
2. The mathematics of chaos and bifurcations
3. The mathematics of history
4. Millennia as the epochs of history between bifurcations
5.
Education
for the chaos millennium
Acknowledgments
Bibliography
In this short lecture I want to discuss
three things:
The first is Chaos, secondly, the Millennium,
and finally and most importantly, Chaos and the Millennium,
and how they go together in an essential way.
Chaos means
the chaos of everyday life, but also,
it means chaos theory, a new branch of
mathematics also known as dynamical systems theory.
The question naturally arises, whether
there is a connection between the mathematical model
of chaos and the chaos of everyday life.
For a long time I rejected this connection,
but now I feel it is justified.
Mathematics has the branches which are standard
in history
books: arithmetic, geometry, dynamics, and so on.
These branches are more or less old.
Algebra and geometry are very old subjects,
coming to us from ancient city-states like Sumer,
Babylon, Indus,
and Egypt.
Dynamics is newer, only three or four hundred
years old. A new branch of mathematics would be a tremendous
novelty, and pose a challenge to the orthodoxy of mathematics
at universities, because for a long
time there has not been a new branch of mathematics.
But now there are. Chaos theory is a new
branch of mathematics, one which has evolved into a mass
movement and major social transformation called
the chaos revolution.
Now I would like to introduce briefly the
five most
essential concepts from chaos theory:
states, trajectories, attractors, basins, and bifurcations.
The most fundamental and important of the
basic notions
of chaos theory is the state space.
This is at the basis of everything and goes back to
geometry. It is an imaginary geometrical model
for a system. Let's say you have a restaurant.
The groceries come in the back door,
and the satisfied customers go out the front door
leaving money behind. Now you count up the supplies and
the money left behind; these counts represent the state
of the restaurant (as an economic system)
at a given instant. But they may also be regarded
as the coordinates of a point in a geometric
space, a model for the system. This geometrical model
is the state space of this example.
Now here enter
the dynamics of the dynamical system:
the system changes. Every minute it
has a slightly different state. So its representative
in the model is a point wandering around in the
state space. It is a moving point which draws a curve
called a trajectory. And this is where mathematicians
put a simple restriction. It is imagined that at every
point (state) in this state space there
is a single vector, which gives the direction and speed
of any trajectory which will ever pass though that state.
Suppose we have such model and we are interested
in the
long-run behavior of the system.
This means that we only want to know, twenty years from
now, for example, if there will be a fortune
to pass down to the next generation or not.
In other words: what is the long-run expectation
of a trajectory in the model? We have to start from
one point and follow the instructions, drawing
the trajectory of the system for a long time.
And when we do that, we come to what is called an
attractor. There are three kinds of attractors:
- a point
attractor, otherwise known as a static
attractor,
a good model for death,
- a periodic
attractor, or oscillation,
a model
for simple life, and
- a chaotic
attractor, a model for more complex life.
The third sort, amazingly,
was discovered only recently
and unexpectedly, thanks to computer graphics.
Attractors are models
for the long-run behavior of an
idealized mathematical model for practically
any kind of a dynamical system subject to this very
rigid restriction that the rules of evolution
don't change in time, a so-called autonomous dynamical system.
It is important
to know that a typical dynamical system
has more than one attractor. Suppose,
for example, that there were two. Then some trajectories
would evolve toward one, while other
trajectories would wind toward the other attractor.
And if one attractor models death and the other
life, it may be very important to know which final state
is in your future! The basin of an attractor
is the piece of the state space filled by trajectories
tending to that attractor. If there are two attractors,
then the state space falls naturally into two pieces:
the basin of death and the basin of life.
Each attractor has a basin, and each basin contains a
single attractor.
Now suppose the rules
do change in time.
Then the attractors and their basins might change.
One kind of attractor can change into another kind.
An attractor (along with its entire basin) might
disappear into the blue, or a new one appear out
of the blue. A basin might suddenly explode or
implode, radically changing in size.
These significant changes in the picture of the state space
determined by the attractors and basins are called
bifurcations. There are three kinds of bifurcations:
- subtle bifurcations, in which an attractor changes type
- catastrophic bifurcations, attractors appear out of (or
disappear into) the blue
- explosive bifurcations, in which
attractors drastically change
size.
It is this aspect of chaos
theory which has been of the
utmost importance in the applications to the sciences,
and to history.
So now let
us look at history from the perspective
of chaos theory.
In the preface
of The Bifurcation Paradigm,
subtitled Interesting Times, futurist and systems
philosopher Erwin Laszlo writes:
There is an old Chinese curse that says
"May
you live in interesting times."
I don't believe anyone would dispute that we
do indeed live in interesting
times. Most interesting. But whether these
times are accursed or blessed is
probably less clear to many people. My own
orientation is that neither is
the case. We are under no dark cloud.
Nor do the heavens smile upon us.
What the world will be like for us and our
progeny is very much up to us.
This belief constitutes the basic underlying premise
upon which our work is predicated.
This century has seen the advent of an era
in which the range of possibilities for life,
for the quality of life, and perhaps even for the
persistence of life on our planet, is very much
if not entirely to be determined
by what we, the human inhabitants of the planet, do.
And what we do, we assert, will be a direct consequence
of what we believe and know P of how
we approach problems and situations...
Some people are pessimistic and others optimistic,
but Laszlo says that he is neither. According to
studies made by the World Future Society,
and published in its monthly magazine, The Future, the
predictions of futurists are right about two
times out of three. So we must take Laszlo seriously.
The World Future Society also surveys to see what
other people think about the future. They
surveyed thousands of people, and found that
two-thirds were more optimistic than pessimistic.
Sociologist Paul Ray has studied three subcultures
in the United States.
These he calls:
- the
Heartlanders (Traditional Stream, dating from about 1870),
- the
Modernists (Modern Stream, from 1920), and
- the
Cultural Creatives (Transmodern Stream, 1970).
He defines these groups according to three different
world views. Heartlanders favor a return to the
past. Modernists champion the secular, economic,
and scientific paradigm of the 20th Century. Cultural
Creatives are bearers of a new world view,
currently evolving in reaction to the shadow side
of the Modern Stream. In a social research survey in 1994,
Ray sampled about two thousand people
in the United States.
He found the proportions of
these three subcultures to be:
- Traditional,
29%, equivalent to 56 million adults of today's US population,
- Modern,
47%, or 88 million adults, and
- Transmodern,
24%, or 44 million adults.
He concludes that we are now at a Great Divide between
the Modern Stream and its Transmodern
sequel, the latter yet to be determined.
In this
essay we are going to take sides with
Ervin Laszlo, the World Future Society, Paul
Ray, and the 44 million Transmoderns. And we are
going to bring the Chaos Revolution into the picture.
At one time people thought that history was continuous
and gradually changing, that nothing
ever happened suddenly. This was the opinion of Leibniz,
who contributed ideas of evolution and
linear progression to history. Leibniz is the coinventor
with Newton of the calculus, and was also
a futurist and historian. Applying the new mathematical
ideas of his time to history, he came up
with his principle of continuity. He was a gradualist.
The
idea of bifurcation (or catastrophe) is a
different idea; an idea of discontinuity, of punctuated
or saltatory change, sudden or miraculous transformation.
Laszlo
concludes his preface:
The issues are burning, the stakes enormous,
the options impressive. These
truly are interesting times. This stands to reason.
We stand in humankind's greatest age of bifurcation.
For him,
bifurcation as an important new word and
mathematical concept. He uses it in place of
the equivalent general terms, major social
transformation, or Great Divide. He sees history as
consisting of flat spots punctuated by major social
transformations or bifurcations. This is like the
evolution of species in the Darwinian sense where
long flat spots are followed by the sudden
emergence of new species, which may be triggered
by collisions with comets. Laszlo is a bifurcationist.
Even more: he says that this current transformation,
our own bifurcation, is the biggest one
in human history. Quite an idea!
Historiography is the study of structure in history.
And now that mathematics has been redefined as the
study of space-time patterns, we could say that
historiography is the mathematics of history.
Jacob Burkhardt of Switzerland was an early
bifurcationist. He analyzed the Italian Renaissance
as a catastrophic bifurcation, and made his reputation
with this theory. It was the first of the
saltatory or discontinuous theories of history,
coming shortly after Leibniz and his principle of
historical continuity.
A controversy developed over this interpretation:
was the Renaissance a sudden, catastrophic
bifurcation, or just a gradual change? The
conservatives in this controversy believed in the
continuity of history. They noted that a century
earlier, Petrarch, Bocaccio, and others had introduced
new ideas in literature which were more or less
characteristic of what was later called Renaissance
Humanist thought. They said that history was continuous,
while Burckhardt and his followers said
that it was discontinuous. This controversy,
still ongoing today, led to a whole school of the
philosophy of history, which came to be known
as historiography.
My idea of historiography consists of the space-time
pattern of world cultural history, the history movie,
superimposed over the space-time pattern of our biosphere,
the biogeographical movie. But the simplest representation
of this combined space-time pattern is a histomap.
In a histomap, geography, which really lives on the
two-dimensional sphere of the earth, is reduced to one
dimension for convenience, so that geographical space
plus time comprise a two-dimensional display in which
to locate space-time events.
A histomap by Edward Hull, from around 1900, may be the
best histomap ever made. In it,
the map of the world is a one-dimensional,
vertical interval about eighteen inches high.
The time span is a horizontal line about four feet long.
It starts in 2500 BC and ends more or less now, with
approximately two inches corresponding to a century.
A later histomap was the first to actually
uses the word histomap. (Sparks, 1931)
A small
piece of Hull's histomap is shown in Figure 1.
The colored streams denote cultures.
They suddenly get wider when there are more people or
more territory. There are thin ones and
thick ones and they change in their relative sizes
as we go down the river of time. Between and
around them are icons from archaeological and
historical records.
A bifurcation is a special feature of chaos theory.
Applied to historiography, the study of the
space-time patterns in history, a bifurcation would be
a significant historical event. When you
look at a histomap, a bifurcation should just
jump out at you.
For example, let's look at Hull's histomap,
in Figure 1.The vertical bar in the center is an
exemplary bifurcation, the conquest of
Alexander the Great. In a decade he conquered most of the
known world. All of the separate strands
of different cultures were united by him personally,
so all the horizontal rivers of culture join that vertical
bar. Upon Alexander's death his empire
disintegrated, so the vertical bar breaks up into more
rivers. This was a double bifurcation.
If world cultural history is punctuated
by quantum leaps, then there must be periods of
gradual change in between the major bifurcations.
These we call epochs. The division of the whole of
history and prehistory into epochs is a
subjective process, and many such divisions, or schemes,
have been proposed. We will now review a few of them.
Ancient skywatchers followed the histomap of the zodiac.
The major bifurcations were
marked by the passage of the vernal equinox from one
zodiacal constellation to another. These
events occur roughly every 2,200 years, and are very
important in all early cultures. (De Santillana, 1969)
Within the recent millennia of the Holocene
Interglacial, here are the dates:
10,500
BC, Leo; 7,900 BC, Cancer; 6,500 BC, Gemini
4,500
BC, Taurus; 1,900 BC, Aries; 100 BC, Pisces
2,500
AD, Aquarius
In Chaos, Gaia, Eros, a scheme of three epochs
is proposed, with major bifurcations:
10,000
BC, the agricultural revolution.
4000
BC, the discovery of the wheel,
1972,
the chaos revolution.
William Irwin Thompson is one of the leading cultural
historians of our time. His approach to
world cultural history, which is both cultural and
mathematical, has a division into four periods
which he calls cultural ecologies. Associated with
each is a characteristic mathematical style, or
mentality. These, with beginning dates, are:
4,000
BC, the riverine cultural ecology, with the arithmetic mentality
500
BC, transcontinental, geometric
1700
AD, oceanic, dynamic
1972,
biospheric, chaotic
Originally, the millennium meant a particular period of
one thousand years, which was to follow a great
bifurcation, according to an apocalyptic doctrine
of the ancient Hebrews. (See Cohn, 1957.)
When the coming of Christ was thought to signal
that event around the year 1 BC, the end
of the millennium (that is, the apocalypse)
was to fall around the year 1000 AD. (See Revelations.)
It did not, but the concept persisted in
groups knows as revolutionary millenarians.
The millenarians believe that the end of this world,
and of historical time, is
at hand. A new world, and a new time,
will be inaugurated, usually through
the agency of a messiah: a saviour or deliverer.
There will be many tribulations and mighty conflicts.
The forces of evil will gather themselves up in a
last bid for victory. But the good will triumph.
The new era P the millennium P will be a time of
peace, plenty and righteousness. (Kumar, 1991; p. 7.)
For the purposes of this essay, I am pirating the
word millennium, and appropriating it as a synonym
for epoch. It has no longer any implication of
a period of one thousand years.
Thus, we now have a catastrophic bifurcation from
the Periodic millennium to the Chaotic
millennium, according to me; from the Oceanic millennium
to the Biospheric millennium according to Thompson;
or from the Modern to the Transmodern, according
to Ray. Whatever you call the epochs,
all agree that we are now in a bifurcation;
one which is the largest so far, according to Laszlo.
In our global development, we have moved
from the ancient arithmetic
mentality to the classical geometrical
mentality to the modernist dynamical
mentality and now finally to the new
chaos dynamical mentality, a mentality that is
based on the new sciences of complexity, on the new art forms
that cross one genre with another, and on the new
multidisciplinary sciences such as Lovelock's
geophysiology that give us a biospheric vision of
our new planetary cultural ecology. (Thompson, 1996; p. 241.)
But in addition to characterizing the dominant
mathematical style of the coming millennium,
the chaos mentality is able to model the great
transformation itself. This is my main message
here: Chaos theory can help us to understand
our unique experience in this special moment of history,
at the dawn of a new millennium. Chaos and the millennium!
Chaos and the millennium is a very important subject.
It is very appropriate for this time, and
it has crucial implications for education in general
and for honors programs in particular. The
school system is where the future is actually created,
and honors programs provide a special
opportunity to teach new paradigms. So here is
something for us to do: the reformation of education
on all its levels, to address the future, and to
create a society which has sustainable future.
New programs
are badly needed which integrate world
cultural history, the new branches of
mathematics, and the willful creation of the future.
Chaos theory must be rescued from the fringes
of academia and brought into the center of our
schools and universities. Curricular reform in our
schools, and integrative programs such as honors
programs in our universities, provide opportunities
for the renewal of our outworn educational system,
and the creation of a viable future.
ACKNOWLEDGEMENTS
A am grateful to my colleagues of the General Evolution
Research Group and the Lindisfarne Association
for the collegial sharing of ideas over the years,
to Courtney Sale Ross for awakening
me to the possibilities of elementary education,
and to Russell Spring for bringing Paul Ray to my attention.
BIBLIOGRAPHY
Abraham, Ralph.
Chaos, Gaia, Eros,
San Francisco: Harper San Francisco, 1994.
De Santillana, Giorgio, and Hertha von Dechend,
Hamlet's Mill,
Boston: Gambit, 1969.
Eisler, Riane,
The Chalice and the Blade: our History, our Future,
Cambridge [Mass.]:
Harper & Row, 1987.
Hull, Edward.
The Wall Chart of World History: with Maps of
the World)s Great Empires and
a Complete Geological Diagram of the Earth. Facsimile ed.
U.S.A.: Dorset Press, 1988.
Laszlo, Ervin.
The Age of Bifurcation: Understanding the Changing World.
Philadelphia: Gordon
and Breach, c1991.
Thom, Rene.
Structural Stability and Morphogenesis;
an Outline of a General Theory of Models.
Transl. D. H. Fowler.
Reading, Mass.:
W. A. Benjamin, 1975.
Redwood City, Calif.:
Addison-Wesley, Advanced Book Program, c1989.
Ray, Paul H.,
The rise of integral culture,
Noetic Sciences Review, Spring 1996: 4-15.
Thompson, William Irwin.
Pacific Shift,
San Francisco: Sierra Club Books, c1985.
Thompson, William Irwin.
Coming into Being:
Artifacts and Texts in the Evolution of Consciousness,
New York: St. Martin)s Press, 1996.
Zeeman, E. C.
Catastrophe Theory: Selected Papers, 1972-1977.
Reading, Mass.:
Addison-Wesley Pub. Co.,
Advanced Book Program, 1977. |