equation describes how interesting quantities (e.g. position and velocity) of the system change, and the configuration space is a representation of all the different possible values those quantities can take. The advantage of this approach should be obvious: it lets us reduce difficult questions about how complicated systems behave to mathematically-tractable questions about tracing a path through a space according to a rule. This powerful modeling tool is the heart of DyST.
Some systems can be modeled by a special class of differential equations: linear differential equations. Intuitively, a system’s behavior can be modeled by a set of linear differential equations if: (1) the behavior of the system is (in a sense that we shall articulate more precisely soon) the sum of the behavior of the parts of the system, and (2) the variables in the model of the system vary with respect to one another at constant rates. (1) should be relatively familiar: it’s just the decompositionalist assumption we discussed back at the end of Chapter Four! This assumption, as we saw, is innocuous in many cases. In the case of a box of gas, for example, we could take the very long and messy differential equation describing how all the trillions of molecules behave together and break it up into a very large collection of equations describing the behavior of individual molecules, and (hopefully) arrive at the very same predictions. There’s no appreciable interaction between individual molecules in a gas, so
- In mathematical jargon, these two conditions are called “additivity” and “degree 1 homogeneity,” respectively. It can be shown that degree 1 homogeneity follows from additivity given some fairly (for our purposes) innocuous assumptions, but it is heuristically useful to consider the two notions separately.
- Ladyman, Lambert, & Wiesner (2011) quite appropriately note that “a lot of heat and very little light” has been generated in philosophical treatments of non-linearity. In particular, they worry about Mainzer (1994)’s claim that “[l]inear thinking and the belief that the whole is only the sum of its parts are evidently obsolete” (p. 1). Ladyman, Lambert, & Wiesner reasonably object that very little has been said about what non-linearity has to do with ontological reductionism, or what precisely is meant by “linear thinking.” It is precisely this sort of murkiness that I am at pains to dispel in the rest of this chapter.
- Fans of Wikipedia style guidelines might call “appreciable” here a “weasel-word.” What counts as an appreciable interaction is, of course, the really difficult question here. Suffice it to say that in practice we’ve found it to be the case that assuming no interaction between the molecules here gives us a model that works for certain purposes. A whole