Showing Text From Page | View Full page with images

the equations provide some kind of abstract representation for the physical effects that actually determine the motion of the planet.

When I have discussed models like the ones in this chapter with other scientists I have however often encountered great confusion about such issues. Perhaps it is because in a simple program it is so easy to see the underlying elements and the rules that govern them. But countless times I have been asked how models based on simple programs can possibly be correct, since even though they may successfully reproduce the behavior of some system, one can plainly see that the system itself does not, for example, actually consist of discrete cells that, say, follow the rules of a cellular automaton.

But the whole point is that all any model is supposed to do—whether it is a cellular automaton, a differential equation, or anything else—is to provide an abstract representation of effects that are important in determining the behavior of a system. And below the level of these effects there is no reason that the model should actually operate like the system itself.

Thus, for example, a cellular automaton can readily be set up to represent the effect of an inhibition on growth at points on the surface of a snowflake where new material has recently been added. But in the cellular automaton this effect is just implemented by some rule for certain configurations of cells—and there is no need for the rule to correspond in any way to the detailed dynamics of water molecules.

So even though there need not be any correspondence between elements in a system and in a model, one might imagine that there must still be some kind of complete correspondence between effects. But the whole point of a model is to have a simplified representation of a system, from which those features in which one is interested can readily be deduced or understood. And the only way to achieve this is to pick out only certain effects that are important, and to ignore all others.

Indeed, in practice, the main challenge in constructing models is precisely to identify which effects are important enough that they have to be kept, and which are not. In some simple situations, it is sometimes possible to set up experiments in which one can essentially isolate each individual effect and explicitly measure its importance. But

From Stephen Wolfram: A New Kind of Science [citation]