The Math of History — War and Peace

Absolute continuity of motion, Tolstoy begins, is not comprehensible to the human mind. Laws of motion become comprehensible only when we examine arbitrarily selected elements of that motion, and a large proportion of human error comes from dividing continuous motion into discontinuous pieces. The ancients' sophism said Achilles could never catch a tortoise he followed, though he travelled ten times as fast: by the time he covers the gap, the tortoise has covered a tenth, and so on forever. The absurd answer came from chopping motion into separate bits, whereas both motions were continuous. Only by admitting the infinitely small, and summing a geometrical progression to infinity, do we solve the problem. A modern branch of mathematics, dealing with the infinitely small, now yields solutions in other complex problems of motion that once seemed insoluble, because it conforms to the chief condition of motion: absolute continuity.

In seeking the laws of historical movement just the same thing happens. The movement of humanity, arising from innumerable arbitrary human wills, is continuous. To understand the laws of that movement is the aim of history. But to arrive at those laws the mind postulates arbitrary and disconnected units. One method is to take an arbitrarily selected series of continuous events and examine it apart from others, though there can be no beginning to any event, for one always flows uninterruptedly from another. The second method is to treat the actions of some one man, a king or a commander, as equivalent to the sum of many individual wills, whereas that sum is never expressed by the activity of a single historic personage. Historical science, trying to draw nearer to truth, takes smaller and smaller units, yet any unit disconnected from others, any assumed beginning, any claim that one personage expresses many wills, is in itself false. Only by taking infinitesimally small units for observation, the differential of history, that is, the individual tendencies of men, and attaining the art of integrating them, finding the sum of these infinitesimals, can we hope to arrive at the laws of history.

The first fifteen years of the nineteenth century in Europe present an extraordinary movement of millions. Men leave their customary pursuits, hasten from one side of Europe to the other, plunder and slaughter one another, triumph and are plunged in despair. What was the cause, by what laws was it governed? Historians lay before us the sayings and doings of a few dozen men in a building in Paris, calling these the Revolution, then a detailed biography of Napoleon and of people favourable or hostile to him, and say: that is why this movement took place. The mind of man refuses this explanation, because a weaker phenomenon is taken as the cause of a stronger. The sum of human wills produced the Revolution and Napoleon, and only the sum of those wills first tolerated and then destroyed them. Every time there have been conquests there have been conquerors, history says. That does not prove that conquerors caused the wars. Whenever the hands of a watch point to ten, neighbouring church bells ring, yet one has no right to assume that the bells are caused by the watch. A locomotive whistle and turning wheels accompany motion, but they are not its cause. Peasants say a cold wind blows in late spring because the oaks are budding; the coincidence is real, the cause is not the buds. To find the cause one must change the point of view and study the laws of steam, of bells, and of wind. History must do the same: leave aside kings, ministers, and generals, and study the common, infinitesimally small elements by which the masses are moved. No one can say how far man may advance that way, but only along that path does the possibility of discovering the laws of history lie.