Equations Don’t “Govern” Reality

The experiences I have had working with scientists has broadened my worldview in several different ways. Pretty much all of those experiences have a common theme. It turns out that I am a pretty naive person. When an authority figure tells me something, I tend to give their opinions weight. For instance, when I take a course and am told that a particular phenomenon is “governed” by an equation, I give that phrasing the weight that seems due. As if somehow the equation is the formal truth, and the phenomenon is just a particular realization of that formal truth.

This notion was assumed throughout most of my mathematical training (excepting courses given by my advisor). Somehow the wave equation, a partial differential equation, decides how a particular wave will behave. Except that it doesn’t. The wave equation gives insight into how a wave should evolve given certain specific assumptions. Any scientist or applied mathematician will read that last sentence and say, “Duh!” But that’s because scientists are used to thinking of these equations as models, rather than as the thing that the model is attempting to mimic.

All equations that are solved to make predictions are models. They don’t govern anything. They are quantitative descriptions of what we think is going on in reality. As such, we should always keep in mind that when we analyze and solve these equations, that we cannot necessarily project from the equations onto reality. I’m being very conservative with that last statement, because the most successful theories are obvious counterexamples to it. For example, relativity predicting the existence of black holes. Indeed, the ability of a model to predict unforeseen phenomena is one useful metric for determining how much to trust a particular model.

Again, I’m not speaking to scientists here, but rather to mathematicians and others who hold to the Platonist view that somehow the mathematical representation of a thing is the “real” thing. Modern physics calls its best representation of reality the Standard Model. It’s hard for me to make my point better than that.