Sarang, in comments, says:
"I should point out that I _don't_ think my objections are of the pedantic-asshole type. As I see it, the macro-regularities follow certain laws and the micro-regularities follow some, apparently entirely different, laws. You can pretend that these sets of laws are 'continuously connected,' but as far as I can tell, they are entirely different, and also as far as I can tell, there's a watershed at say N = 10000 people at which the micro-laws cease to be continuously connected to the macro-laws."
No one is accusing Sarang of asshole pedantry (pederasty is another matter). But let me say what I think Tolstoy's analogy is and isn't doing.
I think Tolstoy is primarily objecting to the idea that particular events happened because Napoleon did one thing and not another. And indeed, Napoleon did have command of an empire, and his whim really did matter. But Tolstoy would say, I think, that Napoleon's control of his empire, and his army, depended on everyone obeying his orders, which in turn depended on their own decisions, their own "will." You can't understand the French invasion of Russia merely by reference to Napoleon's ambitions.
A true account of the French invasion would be much more fine-grained. Why did each soldier choose to obey his orders (or why did some desert, or whatever)? Why did gun-makers make the necessary guns? Why did so many non-French serve? etc. etc.
But anyway, I think continuity of historical theories is actually a red herring. It may be that as you include more and more dimensions in your model, you get discontinuities, and I don't think it matters. The point is to abandon crude, "discrete" accounts of history and embrace the multitude of factors that come into play. In my analogy, to zoom in, to take a close look at each section of coastline, to avoid the easy "because Napoleon said so" answers.
And I think ultimately the upshot is that history is intractable. History is the confluence of so many individual decisions and outcomes that the only true depiction of the world is the world itself. One can only zoom in so far before outstripping one's data and ability to process it. And this is my problem with Tolstoy's stance - I think he overreacts to the "great man" theories of history that repulsed him. In a sense, he was dealing with the same issues that Krugman deals with in, again,
my favorite Krugman essay.
You understand history as well as you can with the tools you have. But remember, Tolstoy was writing at a time when it was becoming feasible to understand things much better than previously. His desire for a
more continuous account of history is entirely understandable and defensible.
And, finally, the literalism of his analogy - sure, he doesn't seem to have been up to speed on calculus (though who am I to criticize?). But
War and Peace is a novel (or an epic, or whatever Tolstoy thought it was). And this isn't a case of Tolstoy making sweeping statements about a discipline he knew nothing about, as with the
Sokal hoax. In the first place, Tolstoy isn't commenting on mathematics, he's commenting on history, which, given the setting of
War and Peace, I think he earned the right to do. Additionally, I think his commentary is at the very least coherent and informed, which is apparently not true of the commentary protested by Sokal.
So I don't think literalism is called for. At the very least, I feel that I got something out of the analogy. It reflects the philosophy that Andrei had embraced during the invasion, and I think it's something we've all thought about in one form or another. I'm actually thrilled to be able to see something the way Tolstoy saw it, even if I can't be sure that I precisely appreciate his point.