Saturday, February 9, 2008

Accuracy, Precision, Description

Today you'll hear a lot of talk about the predictive (or descriptive) power of models -- mathematical models, physical models, usually run by powerful computers. A model is a simulation that describes what a part of the world is like, or can predict what one part of the world will be like in the future (or what it was like in the past). Of course, the description can't be so detailed that it describes the world perfectly -- with no stone left unturned. But this "imperfection" is probably the very reason they are talked about so much. It's generally assumed that if the model can't describe the world in great detail, it mustn't be a good model at all.

But what counts as "great detail"? And what good would models be if they mimicked the world in such great detail that they'd...become the world?

In his book, On Exactitude in Science, the author Jorge Luis Borges quotes a fictional author who tells of an empire that "perfected" their maps (their model making) to the point of absurdity.

"In that Empire, the craft of Cartography attained such Perfection that the Map of a Single province covered the space of an entire City, and the Map of the Empire itself an entire Province. In the course of Time, these Extensive maps were found somehow wanting, and so the College of Cartographers evolved a Map of the Empire that was of the same Scale as the Empire and that coincided with it point for point. Less attentive to the Study of Cartography, succeeding Generations came to judge a map of such Magnitude cumbersome, and, not without Irreverence, they abandoned it to the Rigours of sun and Rain. In the western Deserts, tattered Fragments of the Map are still to be found, Sheltering an occasional Beast or beggar; in the whole Nation, no other relic is left of the Discipline of Geography." (*)

This echoes an exhange in one of Lewis Carroll's books, Sylvie and Bruno Concluded.

"That's another thing we've learned from your Nation,' said Mein Herr, `map-making. But we've carried it much further than you. What do you consider the largest map that would be really useful?'
`About six inches to the mile.'
`Only six inches!' exclaimed Mein Herr. `We very soon got to six yards to the mile. Then we tried a hundred yards to the mile. And then came the grandest idea of all! We actually made a map of the country, on the scale of a mile to the mile!'
`Have you used it much?' I enquired.
`It has never been spread out, yet,' said Mein Herr: `the farmers objected: they said it would cover the whole country, and shut out the sunlight! So we now use the country itself, as its own map, and I assure you it does nearly as well. Now let me ask you another question. What is the smallest world you would care to inhabit?' (*)

Ultimately, the issue is whether the information conveyed by any description is enough for whatever purpose seems to be at hand. But what some critics of models seem to forget is that when they demand an incredibly high standard of description, they border on asking the absurd. Of course, there are the ones that conclude that the description isn't accurate because it's not absolutely precise. (So 2 + 2 doesn't equal some number between 3 and 6?). But a lack of precision doesn't translate into a lack of accuracy, although admittedly we want to have some kind of precision in our descriptions of the world. The statement, "it's 12 o'clock give or take 12 hours" is accurate, but so imprecise that it's not a valuable description at all.

Even in a complex world, a relatively simple description can be useful. It can give you the broad picture, and very rarely do small and seemingly insignificant details matter much. Anyone who muses that you need to be more specific or detailed in order to "get the picture" is asking too much. Creationists do it all the time: "show me precisely how and when humans evolved from their ancestors, or else you can't really claim at all that we're the products of evolution". Or, better yet, "until we can pinpoint exactly how much humans are contributing to global warming, how can we say humans are contributing at all?".

There are times, though, when such small details matter. Modelling the future is a difficult business, and in a world where small things can lead to big changes, it often helps to be thoroughly detailed even about purportedly insignicant things. But there comes a point when the model must, by definition, fall just short of reality (or the reality of what is to come). To abandon the model on that account, though, is foolish -- it still can tell you a lot. There really isn't "only one way to find out". You can find out, more or less; get an good idea, but not the clearest picture.

On that note, time to add my own allegory. I used to debate whether I'd be a good enough visual artist to be an architect. Will I be able to draw my ideas quickly and in the proper detail? Over time I learned that a not-so-perfect drawing is actually the most elegant of all. There is something of character in a representation that doesn't give too much away -- but enough to appreciate what is to be built.

Of course, there are those computer programs that strive to make a design seem exactly as if it's been built. I suppose this was a solution to the problem of the skeptical client who might say, "I won't agree to this design unless I see what it would really look like -- none of these fancy drawings and models work for me". To which the architect then replies, "I can build bigger ones".

1 comment:

Anonymous said...

Very Interesting!
Thank You!