Reading further, pi may have been a bad example. It's considered a mathematical constant, not a physical one (and I was trying to stay away from physical ones because most of them have to do with space-time). And reading from people who actually know what they're talking about, pi would hold in different universes, and also, in entirely different spaces (hyperbolic space, for example). There's also the fun in that pi is a number, but is irrational, and as such, can only be defined by a series expansion, and not by any numerical representation.

(also, pi*radius^2 is area, not diameter)

So if I can't use pi, I can still go after things like the gravitational constant, Avogadro's number, the Planck constant, etc. Things that are based both on empirical models of physical phenomena, as well as coming from the realm of pure reasoning and logic. These can change, and have changed their accepted values over the years, and have been posited to be different under certain circumstances (different universes, incorrect physical models, etc.)

Anyhow. All I was saying is I think there are objects that make the distinction between rational reasoning and empirical observation less cut-and-dry.