HISTORY AS POINTS AND LINES
Manuscript
CONTENTS
PROLOGUE
1. Post hoc ergo propter hoc 15. Invisible walls of events
2. Atomism 16. Generators, configurations, patterns
3. E pluribus unum 17. Law and disorder
4. Alternatives and altercations 18. Conflict
5. Three views of the world 19. Testing the Ariadne’s thread
6. Graphs 20. Probability and energy
7. King Pear 21. Ideas and actions
8. Groups 22. Three World Wars
9. To understand the world 23. The French Revolution
10. From Augustus to Nero 24. The Fall of the Soviet Empire
11. Syntax and semantics 25. The Circuitry of Imperial China
12. Geopolitics of Europe 26. Order-chaos, heat-cold
13. History in the making 27. History and computers
14. Fermentation in wine
barrel
28.
A sunny day in September
and society
CONCLUSION
PROLOGUE
Change
must follow a pattern or leave us unaware of its existence.
G.J.Renier
History and mathematics—can they be put side by side in any
sense?
Well,
it is possible...
...some people dislike the study of history, just as others dislike the study of mathematics... [C. Brinton, J.B.Christopher, R.L.Wolff, A History of Civilization, Vol II, Englewood Cliffs: Prentice Hall, 1956, p.3]
What about history and
chemistry?
History is the most dangerous product developed by the chemistry of the intellect. [Paul Valéry, The collected works of Paul Valéry, Vol. 10, New York: Pantheon Books, 1962., p. 114]Taking history and mathematics, do they have anything to tell each other?
Automatic target recognition, body movements, behavior, mathematical logic, growth and decay, language, human and animal skeletons, grammars, mathematical functions, automata, industrial processes, weave patterns of fabric, molecules, handwriting, spectra, cockroach's legs, human hands, the crust of the earth, genealogy, plots of novels and fairy tales, archaeology, motion of planets, kinship relations, botanical taxonomy, scientific hypotheses, social dominance, language, anatomy, and much more—actually, anything.
So, why not history,
more
captivating than any fairy tale, the greatest and longest novel
ever
written,
full of suspense, tragedy, and hope, with heroes and monsters of
global
proportions?
The comparison with X rays, however vague, is meaningful for us.
We
can examine with X rays a multitude of totally unrelated objects
as
different
as live human hand, Rembrandt’s painting, Egyptian mummy, horse’s
leg,
and the suitcase of an airline passenger. What we see in all cases
is a
black-and-white shadow with very little resemblance of what is
seen on
the surface. It can be easily scanned and reduced to a long
sequence of
numbers stored in computer memory, each presenting the darkness or
brightness
of a certain point. All X ray photos, so to say, are made of the
same
stuff,
although they are derived from strikingly different objects. They
revoke
the image of the shadows on the wall of the Plato’s cave.
What unites all those
heterogeneous
subjects is that they have structure and they are complex. As
viewed
through
the mental imaging apparatus of mathematics, they may look
surprisingly
similar: like points connected with lines.
Not accidentally, the
application
of pattern theory to representing and understanding medical
imaging is
one of the fastest growing. The same principles can be applied to
the
body
of historical narrative, or, rather, its revealed skeleton.
Following this analogy,
we will first try to show how to produce the image of the
skeleton,
and,
secondly, speculate on how to understand what it tells us.
Since our experiment is first of the kind, we cannot promise too
much.
If, however, it will stimulate a historian to repeat it in a more
professional
fashion, we would say that our expectations were exceeded.
The general mindset of
pattern
theory is not alien to modern culture and we believe that the
reader
will
easily recognize its relation to past and current trends, atomism
and
structuralism
in particular. There is a far more general relation, however.
The second half of the twentieth century has been deeply imprinted
by a dichotomy Charles P. Snow attributed to the contemporary
culture.
He saw an impenetrable divide between people living in two
immiscible
media—natural
sciences and liberal arts. His motto about two cultures is still
echoed
by polemic responses. We can mention David Edgerton [
Nature,
389, 221 (1997) ] and John Brockman [The
Third
Culture;
Beyond the Scientific Revolution, New York, Simon &
Schuster, 1995,
see also Web site www.edge.org]. The confluence is the
sign of
our
times.
Charles P. Snow’s
troubling
picture of alienation of two subcultures that could not find the
common
language lost after Lucretius, Leonardo da Vinci and Goethe was
definitely
exaggerated even in his time: Albert Einstein played violin. At
present,
both opposites—digerati and literati—are separated by a very
diffuse
border.
And yet driving along
the
less traveled route from Physics to Poetry, we feel how the
landscape,
vegetation, the color of the skies, and the language of aborigines
changes,
and somewhere between the exit to Psychology and the exits to
Sociology
and next one to History, we arrive at a different world.
Among the major
changes
we would certainly notice the fading presence of mathematics
(statistics
is the last to wither), the growing presence of images and
metaphors,
the
declining power of logic, and the rising spontaneity of
self-expression.
The mathematician calls many things with one name, while the poet
calls
one thing with many names.
The transition
can
hardly be compared with the abyss, chasm, or canyon: it is
gradual, but
as distinct as the change of climate along a meridian.
When we compare two
sketches
of Leonardo da Vinci, figure P-1, symbolizing the two cultures,
they
seem
to sharply contrast. Is there any way we can bridge them? The
simplified
truss bridge in the picture presents a simple pattern of identical
triangles
with joint edges. We can see a high degree of regularity in this
pattern:
the “typical” upright triangles alternate with triangles upside
down.
We
can describe the pattern in a few words of common language, and we
can
extend it in any direction.
When we start
looking
at the world as a collection of patterns, it loses most of its
color,
smell,
weight, texture, beauty, and other material and spiritual
qualities. It
looks like the shadows of X rays. We can code them in sequences of
numbers,
as in digital photography, but what they tell can be expressed in
points
and lines.
For some readers points
and lines may invoke the childhood’s nightmare of a geometry
textbook.
The image of history that is one for all, like the multiplication
table,
could be terrifying. It would be only natural to doubt whether we
can
gain
anything by applying any stern ascetic approach to liberal arts
such as
history, full of life, color, mood, and bias, flapping in the
transient
winds of today’s political weather. Note, however, that numbers
are
rarely
present in a textbook of geometry mostly as page numbers: geometry
is
an
example of mathematics without numbers. The kind of mathematics we
a
trying
to betroth to history is even more liberal: there is neither angle
nor
length.
Whether the marriage can take place, we cannot tell. But with a
mutual
curiosity on both sides, computers are the most powerful marriage
brokers
of our time.
Having mentioned in
this
brief introduction some apparently disconnected terms, we are now
going
to use them as the piers of our bridge symbolizing the profound
unity
of
our world and our knowledge about it.
We started our
exploration
with basic ideas of pattern theory as navigation tools, but
without a
map.
We were looking for a new intellectual passage to familiar waters
of
historical
records. What we found was more than we had expected: the
well-traveled
waters branched into new passages leading to new waterways toward
unknown
horizons. We hope that other seafarers will start where we had to
turn
back.
Our primary goal here
is
to draw attention of new intellectual adventurers to pattern
theory and
provide them with the basic tools that one of the authors
developed.
In first fifteen chapters we browse through some episodes of
history,
share some observations on the use of geometrical imagery in the
works
of historians, and prepare the reader to handling a few basic
mathematical,
physical, and chemical ideas such as binary relation, graph,
transformation,
group, atomism, and principles of thermodynamics. We want to show
that
most of those ideas have always been secretly present in works of
historians
and sociologists. At the same time we want to locate our own place
among
different views of the world.
In Chapter 16 we
present
the basics of the concept of pattern. We formulate the ideas of
generator,
bond couple, regularity, bond relation, transformation, etc.
Next, we add more examples and look into dynamic aspects of
regularity,
structural change, and catalysis. In Chapters 22 to 25 we apply
our
tools
to some large historical developments and formations. In the last
two
chapters
we cast some looks into the future.
We assume that many
readers
are very far from mathematics, physics, and chemistry and we
prefer
images,
illustrations, and metaphors to formalism. Instead of a systematic
layout
of a subject we prefer to draw it as a magic tree riddle where the
observer
is invited to find a bunch of birds hidden in the trees. We
promise to
help catch those birds, but the reader should stay alert.
The following first
chapter,
for example, helps the reader to notice a persistent tendency of
our
mind
to find connections between different things and ideas. The
concept of
connection or relation is one of the fundamental concepts of
mathematics,
but we are going to present it here mostly through storytelling.