HISTORY AS POINTS AND LINES
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
A sunny day in September
must follow a pattern or leave us unaware of its existence.
History and mathematics—can they be put side by side in any
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,
captivating than any fairy tale, the greatest and longest novel
full of suspense, tragedy, and hope, with heroes and monsters of
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
brief introduction some apparently disconnected terms, we are now
to use them as the piers of our bridge symbolizing the profound
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.