Math II (Несколько текстов для зачёта), страница 14

But is this modern view of historical events anything more than current academic fashion? Is there a way of deciding which -- if any -- of these two views is closer to the truth? Intriguingly, a number of scientists are starting to claim that there is. They are talking of an astonishing possibility: of using mathematics to cast light on the causes of history.

TWO TIMES TWO IN AN ARMS RACE

To many, the suggestion that mathematics can be applied to human affairs is absurd. People and events surely do not follow the dictates of an equation to four decimal places. Of course, those making the claims are saying no such thing. What they are saying, however, is that mathematics, if used judiciously, can cast an altogether new light on historical arguments.

While the full implications are still being explored, the first hints of this intriguing possibility emerged more than 70 years ago in the work of English physicist Lewis Fry Richardson.

Eclectic in his interests, Richardson is now recognised as one of the pioneers of multidisciplinary research, one who saw no barriers between the physical and social sciences. Trained as a physicist at Cambridge University, he completed a degree in psychology in his late 40s, and spent the last years of his life investigating how mathematics might shed light on armed conflict.

Richardson also wrote two books based on some of his results. The first, Arms and Insecurity, was an attempt to understand arms races, and how they spiral out of control and lead to war. While Richardson was able to derive some support for his ideas from military expenditure figures of opposing forces before World War I, the predictions of the theory have proved hard to square with subsequent wars.

His second book was less ambitious, but is now emerging as potentially far more important. Entitled The Statistics of Deadly Quarrels, it attempted to bring together data on all wars since 1820.

Among the book's various tables is one listing the number of wars in which deaths exceeded a given figure. For example, he found that between the years of 1820 and 1929 there were 24 conflicts in the world, each of which resulted in more than 30,000 deaths. More accurately, Richardson stated the loss of life in terms of "magnitudes", calculated by converting the raw figure into logarithms, with 30,000 becoming a magnitude of 4.5.

His terminology is suggestive, for it hints at a link with another form of global catastrophe, earthquakes. The parallels go deeper than mere words, however. Plotting a graph of the numbers of wars against the death-toll magnitudes they produced, Richardson made a surprising discovery: they followed a straight line. At its extremes, the line reflected common sense, showing that there have been relatively few wars producing huge death-tolls. Skirmishes resulting in several thousand deaths are far more common. But, remarkably, conflicts between these two extremes also lie on the same straight line.

As such, Richardson's warfare graph bears a striking resemblance to another, more famous graph: the Law of Earthquake Violence. The law was first identified in 1956 by American geophysicist Charles Richter and his colleague, Beno Gutenberg.

Like conflicts, devastating quakes are mercifully rare, while minor tremors occur all the time. But, as Richardson also discovered with conflicts, quakes of intermediate magnitude also lie along the same straight line.

What could be the connection between the two? According to new research the answer lies, surprisingly, in the behaviour of heaps of grain.

If you empty a bag of rice onto a plate, you'll end up with a more or less stable conical heap once the grains have settled. Now slowly start adding more grains, one by one. At first it makes no difference. But some of the falling grains will eventually trigger avalanches down the sides of the heap. And every so often, a single grain will cause a whole side of the heap of rice to collapse.

In 1995, Kim Christensen and her colleagues at Imperial College, London, carried out a detailed study of the precise size and number of these avalanches. The results of the study showed that they too followed the same straight-line law as earthquakes and wars, and that the big ones are rare and the small ones common.

Some scientists believe this is significant. For the behaviour of grains is known to be a manifestation of so-called self-organising criticality (SOC), in which systems teetering on the brink of instability suddenly organise themselves into a more stable state.

There's no telling which of the grains will cause the sudden change, or how big the change will be. All that can be said is that the number of such changes and their magnitude will follow a straightline law.

A growing number of geophysicists suspect that the Richter-Gutenberg law of earthquakes is evidence that the earth's crust is in a critical state; the slightest disturbance is capable of generating an earthquake of any magnitude.

But now some researchers think that Richardson's law of the correlation between the size and frequency of conflicts is also linked to criticality. The implications are intriguing. For, as physicist and writer Mark Buchanan points out in his new book, Ubiquity: The Science of History, Richardson's law would then mean that "the world's political and social fabric tends to be organised on the very edge of instability."

It would also mean that, just as a single grain can bring on any size of avalanche, the ultimate size of a conflict cannot be predicted. Trivial causes could lead to a border skirmish or all-out war.

Such implications seem to strike a chord with some historians, who are starting to describe events as social "earthquakes" that follow the same basic law as their geologic counterparts. Niall Ferguson of Oxford University has said that the war-torn period between 1914 and 1945 "may be likened to the slipping of a continental plate, and to the resultant season of earthquakes". The mathematics of criticality adds new and surprising depth to such metaphors.

It also casts light on the key question of whether it makes any sense to search for the cause of a given historical event, in the hope that this same cause may be identified again one day.

If historical events are the result of society being in some form of critical state, then the answer is no. There is no more chance of being able to identify. the key stimulus unleashing historical upheaval than there is of being able to say which piece of falling rice will trigger a huge avalanche of grain.

Scientists working on this fascinating question are quick to insist that they are not claiming a mathematical proof of Henry Ford's famous dictum that "history is bunk". For a start, the evidence that any aspect of society is in a critical state is still tentative, though the search is on for more.

Even so, Richardson's law is certainly suggestive. His 70-year-old graph has since been updated and refined by a number of researchers. But its basic form remains unchanged, and its message is the same: some aspect of society is constantly teetering on the brink of immense change.

Carlyle's view that history boils down to the actions of a few key people is not so much wrong as incomplete. For the picture now emerging with mathematical clarity is that history is not merely the result of the actions of people. They must also be in the right place at the right time if they are to trigger a revolution rather than raise a few eyebrows.

MATHSEMANTICS RESEARCH

Source: ETC: A Review of General Semantics, Spring2001, Vol. 58 Issue 1, p22, 14p

THE MATHSEMANTIC MONITOR

"Investigators have begun to address in earnest the effect that language has on mathematical development"

RESEARCH NOW SEEMS POISED to start in earnest to fill the great academic gap between English and mathematics.

If you're a regular reader of this series, you'll understand the Mathsemantic Monitor's glee in making this announcement. You know that most of the previous twenty-three pieces have deplored in one way or another the total separation of our two most basic disciplines.

The very first piece told of how reports in English said that students had "read 80,000 books" at a school in New Jersey, whose library held only 9,500 books, so that the 80,000 actually applied to the readings rather than to the books. (1) This kind of numerical displacement is a standard feature of good English. To illustrate the significance of such displacements, the article went on to note a Federal Aviation Administration report that it had handled "about 143 million aircraft" in fiscal 1992. Now what makes this statement remarkable is that the entire U.S. scheduled air-carrier passenger fleet at the time amounted to only about five thousand aircraft. With that in mind, you can see that the words, "143 million aircraft," cloak the fact that for the year any particular aircraft had about one chance in seven of encountering an FAA operational error.(2)

What's really significant here is that the cloaking of the one-in-seven chance per aircraft results not from any misadventure of English or math alone, but only from their combination. We fail to penetrate the cloak not because we're inadequate in either math or English, but because we've studied math and English only separately, each in its own special academic compartment. As a result, this kind of numerical displacement by English has not been brought to our attention. We have no inkling that there's a cloak to remove.

The most recent piece more clearly targeted the academic separation. (3) It contrasted our locally general language, English, with the globally special language, math, which happens to communicate relationships, such as interrelated rates of change, particularly well. Unfortunately the impact of the changes, whether environmental or of other practical concern, can't be appreciated without blending both languages, English and math, a blending ability few people have. This situation so concerned the Mathsemantic Monitor that he promulgated a 30th proposition, a summary to go with the 29 earlier ones featured in his book.(4)

30. Mathsemantic competence requires a reasonable combination of simple math and ordinary English (or whatever ordinary language you speak).

To remedy the situation, he pleaded for academic English-math cooperation. "How about it? Let's start tearing down the barrier."

So why the sudden glee? Surely the world has not changed that much. No, but it has changed a little, and so also has the Mathsemantic Monitor's outlook. "Perhaps," he now imagines smilingly to himself, "I'll get to see some English-math rapprochement in my own lifetime." (5)

Older Studies

Interest in the meanings of mathematics has existed since ancient times. Pythagoras, for example, around 540 B.C., presumably after discovering the "wonderful harmonic progressions in the notes of the musical scale, by finding the relation between the length of a string and the pitch ... saw in numbers the element of all things." For Pythagoreans doing geometry, for example, the number "one came to be identified with the point; two with the line; three with the surface, and four with the solid." However, it didn't stop there. "One was further identified with reason; two with opinion"; and "four with justice." "Five suggested marriage, the union of the first even [=2] with the first genuine odd number [=3]." The "attachment of seven to the maiden goddess Athene," seems to have stemmed from the fact that seven isn't involved with multiplication as much as six, eight, nine, and ten. (6)

A quite different interest in mathematical meanings began twenty-five centuries later with the efforts of child psychologist Jean Piaget. He wanted to know how a child came to develop various abilities, including mathematical ones. He and his associates, notably Barbel Inhelder, studied experimentally the child's "conception of" time, number, space, geometry, speed, and logic, among other topics. The many books produced by Piaget and the Piaget laboratories in Switzerland, starting (in French) in the 1920s, reached a flood several decades later. Their most basic finding in each case, always based on experimental results, was that a child's understanding develops in particular stages. (7)

In a child's developing a conception of relationship, for example, Piaget found three stages. To take a nonmathematical instance, in the first stage (at about age six), a child says simply that a brother is a boy. In the second and clearly intermediate stage (at about age nine) a child says that a brother is, "When there is a boy and another boy," but also says that only "the second brother that comes is called brother." Only in the third stage does the child reach an adult conception, "a brother is a relation, one brother to another." (8)

Despite his early emphasis on a child's language (9), Piaget did not question specifically how language might affect a child's acquisition of mathematics. Nor did his followers. For the most part they regarded language as a natural product of the child's thought, and the child's thoughts as a natural product of the child's stage of development. Their research concentrated on determining what these stages were, when they appeared, and how the child moved from one stage to another.