An Object Lesson in Philosophy and Physics
When computing the moment of inertia, there is a small but important step that most people pass over without even noticing it. It requires a little familiarity with Calculus but if you don’t have that, I’ll keep the mathy symbols very brief, so just some skimming will hopefully still get the point across.
If we take dI (a little bit of inertia) to be d(mr^2) (the formula for rotational inertia) then students are taught that this is r^2 dm. This implicitly treats r^2 as a constant but not m—in some sense, treating m, the mass, as the “thing that’s really there” and r, the radius, is defined only relative to m. Why not use the product rule for derivatives and treat both quantities as if they have “equal reality” and therefore get the same mathematical treatment?
The answer is that the mass really is more real than the radius. The mass really is right there, and the radius is our measurement of a distance from some rotational axis. The distance is also real—it is whatever it is, but it only exists relative to the mass element and the rotational axis.
I find that entirely fine and satisfactory, but it does highlight something subtle and worrying for other contexts. We have to have these human, intuitive ideas about what the physical world is like in order to appropriately model physics with Math. What if we were in some much more strange and unintuitive setting, like subatomic physics or the physics of things at high velocity, where our intuitions about which things are “more real” aren’t going to be up to the task of informing the choice of mathematical model?
If our intuitions just can’t help us to choose among the competing mathematical models, does that mean that our biology has limited our capacity to understand nature?