|Ivars Peterson's MathTrek|
October 29, 2001
"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."
This striking (and perhaps cryptic) passage contrasting the continuous and the discrete starts off the 11th book of the epochal novel War and Peace by Leo Nikolayevich Tolstoy (1828-1920). Tolstoy's tale concerns the tribulations of a group of Russian aristocrats during the turbulent period of Napoleon's campaign in Russia.
To reinforce his point, Tolstoy then refers to the ancient story of Achilles and the tortoise. Achilles could travel 10 times faster than a tortoise that he was following. By the time Achilles covered the distance that separated him from the tortoise, the tortoise would have covered one-tenth of the distance ahead of it. When Achilles had covered that tenth, the tortoise would have covered an additional one-hundredth, and so on. Hence, it would appear that you could come to the absurd conclusion that Achilles would never overtake the tortoise.
"By adopting smaller and smaller elements we only approach a solution of the problem, but never reach it," Tolstoy declared. "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 the solution of the problem."
Building on this analogy, Tolstoy turned to the calculus as a model of how to apprehend history. "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," he wrote.
One such problem is the perceived progress of history. "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," Tolstoy argued.
It's likely that Tolstoy was familiar with the work of Pierre-Simon Laplace (1749-1827), computer scientist Paul M.B. Vitányi of the University of Amsterdam notes in a recent, unpublished paper commenting on Tolstoy's references to mathematics in War and Peace.
The success of Newton's laws of motion made it possible for Laplace to envision a completely transparent, deterministic world in which the entire past and future lay within reach. In principle, everything was predictable, and the finest detail accessible to calculation. You could construct yesterday's or tomorrow's world from what you knew today.
At the same time, Laplace imagined the world as a mechanistic ensemble of moving and colliding particles that by their combined microscopic actions produce macroscopic effects. In assessing the role of probability in understanding such a world, Laplace wrote, "I am particularly concerned to determine the probability of causes and results, as exhibited in events that occur in large numbers, and to investigate the laws according to which that probability approaches a limit in proportion to the repetition of events."
"The investigation is one that deserves the attention of philosophers in showing how in the final analysis there is a regularity underlying the very things that seem to us to pertain entirely to chance, and in unveiling the hidden but constant causes on which that regularity depends," Laplace added.
Echoing Laplace, Tolstoy applied an analogous notion to understanding history. "To study the laws of history we must completely change the subject of our observation, must leave aside kings, ministers, and generals, and study the common, infinitesimally small elements by which the masses are moved," Tolstoy wrote in War and Peace.
From Vitányi's perspective, however, Tolstoy was not really seeking in the calculus and Laplacian thought a usable model of history as much as he was trying to demonstrate the futility of the quest for explanations of wars' causes and outcomes.
"This is not a matter of saying that the future is in the laps of the gods," Vitányi contends. "Rather that it is deterministic and determined precisely, [though] practically and possibly in principle unknowable by humans."
"To the imperfect human mind not all information can be available in a snapshot," he emphasizes, "and so it is reduced to ignorance or at best probabilistic reasoning."
Copyright 2001 by Ivars Peterson
Gillispie, C.C. 1997. Pierre-Simon Laplace, 1749–1827: A Life in Exact Science. Princeton, N.J.: Princeton University Press.
Peterson, I. 1993. Newton's Clock: Chaos in the Solar System. New York: W.H. Freeman.
Tolstoy, L. 1996. War and Peace, G. Gibian, ed. New York: W.W. Norton.
Vitányi, P. Preprint. Tolstoy's Mathematics in War and Peace. Available at http://xxx.lanl.gov/abs/math.HO/0110197.
The full text of Tolstoy's novel War and Peace can be found at http://www.funet.fi/pub/culture/russian/books/Tolstoy/War_and_peace.
Paul M.B. Vitányi has a Web page at http://www.cwi.nl/~paulv/.
Comments are welcome. Please send messages to Ivars Peterson at firstname.lastname@example.org.