# Infinitesimal methods in Tolstoy's War and Peace

Lev Tolstoy, in his novel War and Peace, advocates the use of the methods of calculus when studying history.

See also the book The Hedgehog and the Fox: An Essay on Tolstoy's View of History by Isaiah Berlin, the thesis De pathologie van de veldslag by Eelco Runia, Meulenhoff, Amsterdam, 1995, in Dutch (in particular pp. 192-197 in Chapter 5), and the citations and discussion in Paul Vitányi's essay Tolstoy's Mathematics in ``War and Peace''.

The following fragments of War and Peace were originally taken from the Virginia Tech gopher site. The full text of War and Peace is now available in project Gutenberg.

### Third book, Third part, Chapter 1

Absolute continuity of motion is not comprehensible to the human mind. Laws of motion of any kind become comprehensible to man only when he examines arbitrarily selected elements of that motion; but at the same time, a large proportion of human error comes from the arbitrary division of continuous motion into discontinuous elements. There is a well known, so-called sophism of the ancients consisting in this, that Achilles could never catch up with a tortoise he was following, in spite of the fact that he traveled ten times as fast as the tortoise. By the time Achilles has covered the distance that separated him from the tortoise, the tortoise has covered one tenth of that distance ahead of him: when Achilles has covered that tenth, the tortoise has covered another one hundredth, and so on forever. This problem seemed to the ancients insoluble. The absurd answer (that Achilles could never overtake the tortoise) resulted from this: that motion was arbitrarily divided into discontinuous elements, whereas the motion both of Achilles and of the tortoise was continuous.
By adopting smaller and smaller elements of motion we only approach a solution of the problem, but never reach it. Only when we have admitted the conception of the infinitely small, and the resulting geometrical progression with a common ratio of one tenth, and have found the sum of this progression to infinity, do we reach a solution of the problem.
A modern branch of mathematics having achieved the art of dealing with the infinitely small can now yield solutions in other more complex problems of motion which used to appear insoluble.
This modern branch of mathematics, unknown to the ancients, when dealing with problems of motion admits the conception of the infinitely small, and so conforms to the chief condition of motion (absolute continuity) and thereby corrects the inevitable error which the human mind cannot avoid when it deals with separate elements of motion instead of examining continuous motion.
In seeking the laws of historical movement just the same thing happens. The movement of humanity, arising as it does from innumerable arbitrary human wills, is continuous.
To understand the laws of this continuous movement is the aim of history. But to arrive at these laws, resulting from the sum of all those human wills, man's mind postulates arbitrary and disconnected units. The first method of history is to take an arbitrarily selected series of continuous events and examine it apart from others, though there is and can be no beginning to any event, for one event always flows uninterruptedly from another.
The second method is to consider the actions of some one man- a king or a commander- as equivalent to the sum of many individual wills; whereas the sum of individual wills is never expressed by the activity of a single historic personage.
Historical science in its endeavor to draw nearer to truth continually takes smaller and smaller units for examination. But however small the units it takes, we feel that to take any unit disconnected from others, or to assume a beginning of any phenomenon, or to say that the will of many men is expressed by the actions of any one historic personage, is in itself false.
It needs no critical exertion to reduce utterly to dust any deductions drawn from history. It is merely necessary to select some larger or smaller unit as the subject of observation- as criticism has every right to do, seeing that whatever unit history observes must always be arbitrarily selected.
Only by taking infinitesimally small units for observation (the differential of history, that is, the individual tendencies of men) and attaining to the art of integrating them (that is, 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 of people. 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, and for some years the whole course of life is altered and presents an intensive movement which first increases and then slackens. What was the cause of this movement, by what laws was it governed? asks the mind of man.
The historians, replying to this question, lay before us the sayings and doings of a few dozen men in a building in the city of Paris, calling these sayings and doings "the Revolution"; then they give a detailed biography of Napoleon and of certain people favorable or hostile to him; tell of the influence some of these people had on others, and say: that is why this movement took place and those are its laws.
But the mind of man not only refuses to believe this explanation, but plainly says that this method of explanation is fallacious, because in it 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.
"But every time there have been conquests there have been conquerors; every time there has been a revolution in any state there have been great men," says history. And, indeed, human reason replies: every time conquerors appear there have been wars, but this does not prove that the conquerors caused the wars and that it is possible to find the laws of a war in the personal activity of a single man. Whenever I look at my watch and its hands point to ten, I hear the bells of the neighboring church; but because the bells begin to ring when the hands of the clock reach ten, I have no right to assume that the movement of the bells is caused by the position of the hands of the watch.
Whenever I see the movement of a locomotive I hear the whistle and see the valves opening and wheels turning; but I have no right to conclude that the whistling and the turning of wheels are the cause of the movement of the engine.
The peasants say that a cold wind blows in late spring because the oaks are budding, and really every spring cold winds do blow when the oak is budding. But though I do not know what causes the cold winds to blow when the oak buds unfold, I cannot agree with the peasants that the unfolding of the oak buds is the cause of the cold wind, for the force of the wind is beyond the influence of the buds. I see only a coincidence of occurrences such as happens with all the phenomena of life, and I see that however much and however carefully I observe the hands of the watch, and the valves and wheels of the engine, and the oak, I shall not discover the cause of the bells ringing, the engine moving, or of the winds of spring. To that I must entirely change my point of view and study the laws of the movement of steam, of the bells, and of the wind. History must do the same. And attempts in this direction have already been made.
To study the laws of history we must completely change the subject of our observation, must leave aside kings, ministers, and generals, and the common, infinitesimally small elements by which the masses are moved. No one can say in how far it is possible for man to advance in this way toward an understanding of the laws of history; but it is evident that only along that path does the possibility of discovering the laws of history lie, and that as yet not a millionth part as much mental effort has been applied in this direction by historians as has been devoted to describing the actions of various kings, commanders, and ministers and propounding the historians' own reflections concerning these actions.

### Fourth book, Second part, Chapter 8

A countless number of free forces (for nowhere is man freer than during a battle, where it is a question of life and death) influence the course taken by the fight, and that course never can be known in advance and never coincides with the direction of any one force.
If many simultaneously and variously directed forces act on a given body, the direction of its motion cannot coincide with any one of those forces, but will always be a mean- what in mechanics is represented by the diagonal of a parallelogram of forces.
If in the descriptions given by historians, especially French ones, we find their wars and battles carried out in accordance with previously formed plans, the only conclusion to be drawn is that those descriptions are false.

### Second epilogue, Chapter 11

History examines the manifestations of man's free will in connection with the external world in time and in dependence on cause, that is, it defines this freedom by the laws of reason, and so history is a science only in so far as this free will is defined by those laws.
The recognition of man's free will as something capable of influencing historical events, that is, as not subject to laws, is the same for history as the recognition of a free force moving the heavenly bodies would be for astronomy.
That assumption would destroy the possibility of the existence of laws, that is, of any science whatever. If there is even a single body moving freely, then the laws of Kepler and Newton are negatived and no conception of the movement of the heavenly bodies any longer exists. If any single action is due to free will, then not a single historical law can exist, nor any conception of historical events.
For history, lines exist of the movement of human wills, one end of which is hidden in the unknown but at the other end of which a consciousness of man's will in the present moves in space, time, and dependence on cause.
The more this field of motion spreads out before our eyes, the more evident are the laws of that movement. To discover and define those laws is the problem of history.
From the standpoint from which the science of history now regards its subject on the path it now follows, seeking the causes of events in man's freewill, a scientific enunciation of those laws is impossible, for however man's free will may be restricted, as soon as we recognize it as a force not subject to law, the existence of law becomes impossible.
Only by reducing this element of free will to the infinitesimal, that is, by regarding it as an infinitely small quantity, can we convince ourselves of the absolute inaccessibility of the causes, and then instead of seeking causes, history will take the discovery of laws as its problem.
The search for these laws has long been begun and the new methods of thought which history must adopt are being worked out simultaneously with the self-destruction toward which- ever dissecting and dissecting the causes of phenomena- the old method of history is moving.
All human sciences have traveled along that path. Arriving at infinitesimals, mathematics, the most exact of sciences, abandons the process of analysis and enters on the new process of the integration of unknown, infinitely small, quantities. Abandoning the conception of cause, mathematics seeks law, that is, the property common to all unknown, infinitely small, elements.
In another form but along the same path of reflection the other sciences have proceeded. When Newton enunciated the law of gravity he did not say that the sun or the earth had a property of attraction; he said that all bodies from the largest to the smallest have the property of attracting one another, that is, leaving aside the question of the cause of the movement of the bodies, he expressed the property common to all bodies from the infinitely large to the infinitely small. The same is done by the natural sciences: leaving aside the question of cause, they seek for laws. History stands on the same path. And if history has for its object the study of the movement of the nations and of humanity and not the narration of episodes in the lives of individuals, it too, setting aside the conception of cause, should seek the laws common to all the inseparably interconnected infinitesimal elements of free will.