Yuri Tarnopolsky and Ulf Grenander



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History as points and lines

  Foreword (2006)





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



                                Change must follow a pattern or leave us unaware of its existence.

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?
        History is full of arguments, diverging opinions, and contradicting evaluations. Mathematical proof is either correct or not: a proved theorem is proved for everybody. Since ancient times history used to be a captivating reading. Mathematics is a patented source of headache. History seems to be of no immediate practical value—a dynamic society buries the dead and goes on. Mathematics penetrates and impregnates all modern technology and sciences—all but history. History is art, with Clio as its muse, and mathematics is science.
        Nevertheless, the serious answer is a firm “yes.” Since the four volumes of Social and Cultural Dynamics by Pitirim Sorokin, full of numbers derived from history, scores of textbooks and papers on quantitative methods in history have appeared. There are sites on World Wide Web on the use of computers in history, college courses, and software for historians, and scientific societies.
        More confusing questions arise: what is mathematics in the era of computers and what is history in the era of sociology? What is history—facts? What is mathematics—numbers? Furthermore, what is truth? With so much of confusion, isn’t this book an experiment ? la Doctor Moreau in creating yet another chimera?
        Written by a mathematician and a chemist, this book is certainly an experiment. It is definitely neither about history of mathematics nor about chemistry nor about numbers and calculations.
We are not going to do any historical research, for which we, due to our backgrounds, are not equipped. Our sources on history are mostly selected popular books and our knowledge of the subject, except for the events of the twentieth century that we personally witnessed, is second hand. We offer neither interpretation, nor explanation, nor evaluation. Neither do we want to “open eyes” nor to offer “the real thing.” At this point we are not much interested even in the historical truth itself for the reasons which will be explained.
        The purpose of the experiment is to place history on the examination table of a certain apparatus, like an X ray machine or CAT scan, or PET scan, only this apparatus is mental. It is called pattern theory, a new area of mathematics that was developed by one of the authors and by accident mesmerized the other. We want to take a look at history not because we want to see any hidden defects, but because we can place under scrutiny of pattern theory literally everything.
Here is an excerpt from the list of subjects pattern theory considers:

        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.

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