You're Not The Only One Who Can Play This Game, Tolstoy
How about this for an analogy? I draw this idea from Neal Stephenson's Mother Earth Mother Board:
"The basic problem of slack is akin to a famous question underlying the mathematical field of fractals: How long is the coastline of Great Britain? If I take a wall map of the isle and measure it with a ruler and multiply by the map's scale, I'll get one figure. If I do the same thing using a set of large-scale ordnance survey maps, I'll get a much higher figure because those maps will show zigs and zags in the coastline that are polished to straight lines on the wall map. But if I went all the way around the coast with a tape measure, I'd pick up even smaller variations and get an even larger number. If I did it with calipers, the number would be larger still. This process can be repeated more or less indefinitely, and so it is impossible to answer the original question straightforwardly. The length of the coastline of Great Britain must be defined in terms of fractal geometry."
So how about this: asking why some historical event happened is like asking what the length of Great Britain's coastline is. It is not a meaningless question, but it has no unique answer - the precise answer will depend on the frame of reference, how "zoomed in" you are to the map of Great Britain. And any answer is at best an approximation, because one cannot zoom in infinitely far. Discreteness will always remain.
And in fact, there is a tradeoff between precision and feasibility. If you simply connect points on Great Britain's coastline with mile-long segments and add up their lengths, you are missing a lot of inlets and protuberances, but you have a relatively easy task. This sort of tradeoff is one of the themes of my own favorite Krugman essay, The Fall and Rise of Development Economics.
"The basic problem of slack is akin to a famous question underlying the mathematical field of fractals: How long is the coastline of Great Britain? If I take a wall map of the isle and measure it with a ruler and multiply by the map's scale, I'll get one figure. If I do the same thing using a set of large-scale ordnance survey maps, I'll get a much higher figure because those maps will show zigs and zags in the coastline that are polished to straight lines on the wall map. But if I went all the way around the coast with a tape measure, I'd pick up even smaller variations and get an even larger number. If I did it with calipers, the number would be larger still. This process can be repeated more or less indefinitely, and so it is impossible to answer the original question straightforwardly. The length of the coastline of Great Britain must be defined in terms of fractal geometry."
So how about this: asking why some historical event happened is like asking what the length of Great Britain's coastline is. It is not a meaningless question, but it has no unique answer - the precise answer will depend on the frame of reference, how "zoomed in" you are to the map of Great Britain. And any answer is at best an approximation, because one cannot zoom in infinitely far. Discreteness will always remain.
And in fact, there is a tradeoff between precision and feasibility. If you simply connect points on Great Britain's coastline with mile-long segments and add up their lengths, you are missing a lot of inlets and protuberances, but you have a relatively easy task. This sort of tradeoff is one of the themes of my own favorite Krugman essay, The Fall and Rise of Development Economics.
posted by James | 3:13 PM
9 Comments:
Well, the coastline isn't _really_ a fractal, it's only approximately fractal at some range of lengths.
To work, the fractal analogy would require that history as seen as a theory of interacting states should look "like" history seen as a theory of interacting people. It doesn't, particularly; the assumption that it does leads to the kind of silliness that personifies states and treats them as having sharply defined characters.
Not sure I follow you.
1. Even if Great Britain's coastline is not a fractal, it behaves like a fractal over some range, right? And that's enough, I think, for my analogy to work. It means that there is more than one true answer to the question, "How long is Great Britain's coastline?"
2. It's your second point that I really don't follow. So I take it that as you "zoom in" on Great Britain's coastline, it gets monotonically (and, more to the point, continuously) longer, so that each increment resembles the last increment. Your point is that our theories of history are not continuous - they are qualitatively different, not just incrementally different. Or at least, I take it that is your point.
But to my way of thinking, that is an artifact of the discrete ways in which we look at history. If you actually vary "how much detail do we take into account" continuously, you should end up with historical explanations that shade smoothly into each other.
But maybe I'm missing your point. Also, I think there are other salient dimensions besides level of detail, so I'm prepared to take my analogy less literally than I have up to this point.
I should also note that the length of a fractal is usually well-defined. Fractals can usually be generated as limits of well-defined -- nonfractal -- series, the length of any element of the series is well-defined (possibly infinite), and the limit is typically well-defined as well. Either the lengths converge to something finite or they go to infinity, in which case the fractal is infinitely long.
Well, you could perform a series of continuous transformations that morphed an object into something totally different -- a sphere into a disc, for instance -- and neither of those objects is remotely fractal, or like the other. You can claim that your coarse-grained theory looks like your fine-grained theory because they map smoothly into one another, but that may or may not be useful.
Of course, it might also be the case that your final shape is topologically different from your initial shape (e.g. if you're zooming in on a donut), in which case no about fine-graining will save you.
As for how this applies to politics. Your micro-theory is a bunch of interacting people. Model that as a graph, with two people connected by an edge if they know each other. Everyone knows everyone else so this is a completely connected graph. Now consider N villages; now you're clearly no longer completely connected, and your graph probably has a different topology. Therefore, your theory no longer looks the same, and there are no continuous mappings -- at some point, the graph ceases to be completely connected, and that's a discrete change that can't be decomposed into continuous changes.
I should say that in general mathematicians don't believe in "continuities" except insofar as they're limits of discretizations. This has been the case since Berkeley showed that the infinitesimal is meaningless.
I was the worst fucking math student.
But whatever - I did not mean the analogy to be robust against this level of literalism. I think it is sufficiently fruitful, at least for me, that I am willing to overlook donuts and the like.
The one context in which you're willing to overlook donuts.
So in Newton's calculus, are there "infinitesimals"? I'm a little surprised it took so long to get rid of them, if they are useless.
Yes, Newton and Leibniz were horribly confused and it took like 150 years to fix up the whole question of infinitesimals. In the end, it was (more or less) decided that they had to do with the limits of finite sequences. See any real analysis text for further details.
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.
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