|Tolstoy's famous novel about...well, about war and peace (!) contains long passages explaining an analogy he makes between history and calculus. In particular, he argues that we should view history as being made up of the infinitesimal contributions of individuals integrated together (in the sense of the definite integral of calculus rather than just the standard English meaning of the word). For instance,
|(quoted from War and Peace)|
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.
In his article "Tolstoy's Integration Metaphor" (American Mathematical Monthly (112) 2005, pp 631-638), Stephen T. Ahearn argues that Tolstoy really believed that such a view was necessary (though not necessarily possible to actually achieve). Among his apparent motivations was a desire to see the success of Napoleon's empire as a consequence of the contributions of millions of people rather than solely Napoleon's achievement. This would mean that although the novel itself is obviously a work of fiction, the mathematical component is not intended to be fictional but rather a serious philosophical argument embedded in the story.
For more information and relevant quotes from the novel, see
Tom Koornwinder's web page.
the math comes in briefly, and then Tolstoy moves on.
But it shines like a gem.
As a devoted science and math student, and a fledgling literature student, I took on this magnum opus of a novel as a senior in high school, with no idea what I was getting into. Reading it turned out to be one of the most enlightening experiences of my short life, and this gem of a passage was what allowed me to begin to grasp Tolstoy's philosophy, if I can be so bold as to say I have begun to grasp it, even now. More than anything, I was surprised by Tolstoy's mathematical and scientific understanding, and by the genius of his suggesting that these ideas should be applied to the pursuit of knowledge and understanding in other areas of study.
All of that said, I thank you so much for appreciating this wonderful little underrated passage of a great work of literature. There's nothing more gratifying than learning that others appreciate the same small pleasures as oneself. Keep bridging the gap between the humanities and the sciences!
My favorite book ever.