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,
De pathologie van de veldslag
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
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
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
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.
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