The Information trap: tempted by the devil in the details

Natural History, Sept, 2004 by Neil deGrasse Tyson

Most people assume that the more information you have about something, the better you understand it. Up to a point, that's usually true. When you look at this page from across the room, you can see it's in a magazine, but you can't make out the words. Get closer, and you'll be able to read the article. If you put your nose right up against the page, though, your understanding of the article's contents will not improve. You may get more visual detail, but by being so close you'll sacrifice crucial information--whole words, entire sentences, complete paragraphs. The old story about the blind men and the elephant makes the same point: if you stand a few inches away and fixate on the bard, pointed projections, the long rubbery hose, the thick, wrinkled posts, and the dangling rope with a tassel on the end that you quickly learn not to pull, you won't be able to tell much about the animal as a whole.

One of the challenges of scientific inquiry is knowing when to step back--and how far back to step--and when to move in close. In some contexts, approximation brings clarity; in others it leads to oversimplification. A raft of complications sometimes point to true complexity and sometimes just clutter up the picture. If you want to know the overall properties of an ensemble of molecules under various states of pressure and temperature, for instance, it's irrelevant and sometimes downright misleading to pay attention to what individual molecules are doing. A single particle cannot have a temperature, because the very concept of temperature addresses the average motion of all the molecules in the group. In biochemistry, by contrast, you understand next to nothing unless you pay attention to how one molecule interacts with another.

Let me put the issue this way: When does a measurement, an observation, or simply a map have the right amount of detail?

In 1967 Benoit B. Mandelbrot, a mathematician now at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York, and also at Yale University, posed a question in the journal Science: "How long is the coast of Britain?" A simple question with a simple answer, you might expect. But the answer is deeper than anyone had imagined.

Explorers and cartographers have been mapping coastlines for centuries. The earliest drawings depict the continents as having crude, funny-looking boundaries; today's high-resolution maps, enabled by satellites, are worlds away in precision. To begin to answer Mandelbrot's question, however, all you need is a handy world atlas and a spool of string. Unwind the string along the perimeter of Britain, from Dunnet Head down to Lizard Point, making sure you go into all the bays and headlands. Then unfurl the string, compare its length to the scale on the map, and voila! you've measured the island's coastline.

Wanting to spot-check your work, you get hold of a more detailed Ordnance Survey map, scaled at, say, two and a half inches to the mile, as opposed to the kind of map that shows all of Britain on a single panel. Now there are inlets and spits and promontories that you'll have to trace with your string; the variations are small, but there are lots of them. You find that the O.S. map shows the coastline to be longer than the atlas did.

So which measurement is correct? Surely it's the one based on the more detailed map. Yet you could have chosen a map that has even more detail-one that shows every boulder that sits at the base of every cliff. Cartographers usually ignore rocks on a map, unless they're the size of Gibraltar. Well, I guess you'll just have to walk the coastline of Britain yourself if you really want to measure it accurately--and you'd better carry a very long string so that you can run it around every nook and cranny. But you'll still be leaving out some pebbles, not to mention the rivulets of water trickling among the grains of sand.

Where does all this end? Each time you measure it, the coastline gets longer and longer. If you take into account the boundaries of molecules, atoms, subatomic particles, will the coastline prove to be infinitely long? Not exactly. Mandelbrot would say "undefinable." Maybe we need the help of another dimension to rethink the problem. Perhaps the concept of one-dimensional length is simply ill suited for convoluted coastlines.

Playing out Mandelbrot's mental exercise involved a newly synthesized field of mathematics, based on fractional--or fractal (from the Latin fractus, "broken")--dimensions rather than the one, two, and three dimensions of classic Euclidean geometry. The ordinary concepts of dimension, Mandelbrot argued, are just too simplistic to characterize the complexity of coastlines. Turns out, fractals are ideally suited for describing "self-similar" patterns, which look much the same at different scales. Broccoli, ferns, and snowflakes are good examples from the natural world, but only certain computer-generated, indefinitely repeating structures can produce the ideal fractal, in which the shape of the macro object is made up of smaller versions of the same shape or pattern, which are in turn formed from even more miniature versions of the very same thing, and so on indefinitely.