Continuity and History
And while I'm quoting amazing passages, I love this one from volume III, part 3 of War and Peace by Leo Tolstoy (as translated by Richard Pevear and Larissa Volokhonsky):
A well-known so-called sophism of the ancients posits that Achilles can never overtake a tortoise that is walking ahead of him, even though Achilles walks ten times faster than the tortoise: while Achilles covers the distance that separates him from the tortoise, the tortoise will get ahead of him by one tenth of that distance; Achilles covers that one tenth, the tortoise gets ahead by one hundredth, and so on to infinity. The ancients considered this problem insoluble. The nonsensical conclusion (that Achilles will never overtake the tortoise) resulted only from the fact that discrete units of movement were introduced arbitrarily, while the movement of both Achilles and the tortoise was continuous.
By taking smaller and smaller units of movement, we only approach the solution of the problem, but never reach it. Only by allowing for an infinitesimal quantity and the ascending progression from that up to one tenth, and by taking the sum of that geometrical progression, do we arrive at the solution of the problem. A new branch of mathematics, having attained to the art of dealing with infinitesimal quantities in other, more complex problems of movement as well, now gives answers to questions that used to seem insoluble.
This new branch of mathematics, unknown to the ancients, in examining questions of movement, allows for infinitesimal quantities, that is, such as restore the main condition of movement (absolute continuity), and thereby corrects the inevitable error that human reason cannot help committing when it examines discrete units of movement instead of continuous movement.
The same thing happens in the search for the laws of historical movement.
The movement of mankind, proceeding from a countless number of human wills, occurs continuously.
To comprehend the laws of this movement is the goal of history. But in order to comprehend the laws of the continuous movement of the sum of all individual wills, human reason allows for arbitrary, discrete units. The first method of history consists in taking an arbitrary series of continuous events and examining it separately from others, whereas there is not and cannot be a beginning to any event, but one event always continuously follows another. The second method consists in examining the actions of one person, a king, a commander, as the sum of individual wills, whereas the sum of individual wills is never expressed in the activity of one historical person.
Historical science in its movement always takes ever smaller units for examination, and in this way strives to approach the truth. But however small the units that history takes, we feel that allowing for a unit that is separate from another, allowing for the beginning of some phenomenon, and allowing for the notion that all individual wills are expressed in the actions of one historical person, is false in itself.
Any conclusion of historical science, without the least effort on the part of criticism, falls apart like dust, leaving nothing behind, only as a result of the fact that criticism selects as an object for observation a larger or smaller discrete unit, which it always has the right to do, because any chosen historical unit is always arbitrary.
Only by admitting an infinitesimal unit for observation - a differential of history, that is, the uniform strivings of people - and attaining to the art of integrating them (taking the sums of these infinitesimal quantities) can we hope to comprehend the laws of history.
A well-known so-called sophism of the ancients posits that Achilles can never overtake a tortoise that is walking ahead of him, even though Achilles walks ten times faster than the tortoise: while Achilles covers the distance that separates him from the tortoise, the tortoise will get ahead of him by one tenth of that distance; Achilles covers that one tenth, the tortoise gets ahead by one hundredth, and so on to infinity. The ancients considered this problem insoluble. The nonsensical conclusion (that Achilles will never overtake the tortoise) resulted only from the fact that discrete units of movement were introduced arbitrarily, while the movement of both Achilles and the tortoise was continuous.
By taking smaller and smaller units of movement, we only approach the solution of the problem, but never reach it. Only by allowing for an infinitesimal quantity and the ascending progression from that up to one tenth, and by taking the sum of that geometrical progression, do we arrive at the solution of the problem. A new branch of mathematics, having attained to the art of dealing with infinitesimal quantities in other, more complex problems of movement as well, now gives answers to questions that used to seem insoluble.
This new branch of mathematics, unknown to the ancients, in examining questions of movement, allows for infinitesimal quantities, that is, such as restore the main condition of movement (absolute continuity), and thereby corrects the inevitable error that human reason cannot help committing when it examines discrete units of movement instead of continuous movement.
The same thing happens in the search for the laws of historical movement.
The movement of mankind, proceeding from a countless number of human wills, occurs continuously.
To comprehend the laws of this movement is the goal of history. But in order to comprehend the laws of the continuous movement of the sum of all individual wills, human reason allows for arbitrary, discrete units. The first method of history consists in taking an arbitrary series of continuous events and examining it separately from others, whereas there is not and cannot be a beginning to any event, but one event always continuously follows another. The second method consists in examining the actions of one person, a king, a commander, as the sum of individual wills, whereas the sum of individual wills is never expressed in the activity of one historical person.
Historical science in its movement always takes ever smaller units for examination, and in this way strives to approach the truth. But however small the units that history takes, we feel that allowing for a unit that is separate from another, allowing for the beginning of some phenomenon, and allowing for the notion that all individual wills are expressed in the actions of one historical person, is false in itself.
Any conclusion of historical science, without the least effort on the part of criticism, falls apart like dust, leaving nothing behind, only as a result of the fact that criticism selects as an object for observation a larger or smaller discrete unit, which it always has the right to do, because any chosen historical unit is always arbitrary.
Only by admitting an infinitesimal unit for observation - a differential of history, that is, the uniform strivings of people - and attaining to the art of integrating them (taking the sums of these infinitesimal quantities) can we hope to comprehend the laws of history.
posted by James | 1:43 PM
6 Comments:
I don't know, I like it a lot. I think the Achilles-tortoise false paradox is an elegant demonstration of the power of math, and a good analogy for Tolstoy's view of history, at least as otherwise expressed in War and Peace.
I was inclined to say something similar but less vigorous. I mean, there are useful models, and there are stupid ones. Leo has a point, but its profundity does not substantiate his imagination.
That said, I need to read me some Tolstoy.
To expand a little. You don't need continuity to resolve Zeno's paradox, only convergent (i.e. Cauchy) series, which are pretty unrelated. The Greeks understood continuity in geometry (Euclid Bk 2-3 anticipate Dedekind cuts), and Archimedes understood infinite series (see Wikipedia "series"). Also, the idea of an infinitesimal was shown to be incoherent in the 18th cent and most of 19th cent math was devoted to the work of understanding calculus without infinitesimals.
Tolstoy's analogy between history and calculus is remarkably incoherent. Doing calculus with a finite mesh gives you perfectly good answers except when the function is pathological, in which case you can't do calculus at all.
Also I guess I think of Tolstoy's theory of history in W/P as being on the same plane as Derrida's -- as being a bunch of random nonsense that seems almost to cohere because it's embedded in an old-fashioned story.
Please not to use "Tolstoy" and "Derrida" in the same sentence.
Don't Derride him!
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