Many physical models, like gravity, have the nice property of stably approximating reality. Perturbing the positions of planets by one millimeter doesn’t explode the Solar System the next second.
The stability of orbits when perturbing the position of planets is a nice property (from our perspective of not wanting to crash into the sun) of the physical system of gravity. The fact that our model of gravity explains this stability is a nice property of the model, just as, if the physical system did not have the stability, not predicting stability that is not there would be a nice property of the model. As Tarski would say, if orbits are stable, we want our model to predict stable orbits, if orbits are unstable, we want our model to predict unstable orbits.
I didn’t mean the stability of planetary systems as t goes to infinity—this is a very non-trivial problem, AFAIK unsolved yet. I only meant that, if we slightly perturb the initial conditions at t=0, the outcome at t=epsilon likely won’t jump around discontinuously.
I did not intend to dispute the stability of orbits. I mean to point out that the stability is a nice property of the territory, and it is only a nice property of the map because it is a property of the territory. Generally, we should not let our desire that maps have nice mathematical properties override our desire that the map reflects the territory. If the territory has discontinuities at corner cases, the map should reflect it, even though we like continuous functions.
More to the point, there are a variety of systems where the territory displays a degree of sensitive dependence on initial conditions at some scale that makes a stable map impossible.
In fact, on astronomical time scales the dynamics of the solar system (or generally, any multi-body gravitational system) displays such behaviors.
The stability of orbits when perturbing the position of planets is a nice property (from our perspective of not wanting to crash into the sun) of the physical system of gravity. The fact that our model of gravity explains this stability is a nice property of the model, just as, if the physical system did not have the stability, not predicting stability that is not there would be a nice property of the model. As Tarski would say, if orbits are stable, we want our model to predict stable orbits, if orbits are unstable, we want our model to predict unstable orbits.
I didn’t mean the stability of planetary systems as t goes to infinity—this is a very non-trivial problem, AFAIK unsolved yet. I only meant that, if we slightly perturb the initial conditions at t=0, the outcome at t=epsilon likely won’t jump around discontinuously.
I did not intend to dispute the stability of orbits. I mean to point out that the stability is a nice property of the territory, and it is only a nice property of the map because it is a property of the territory. Generally, we should not let our desire that maps have nice mathematical properties override our desire that the map reflects the territory. If the territory has discontinuities at corner cases, the map should reflect it, even though we like continuous functions.
More to the point, there are a variety of systems where the territory displays a degree of sensitive dependence on initial conditions at some scale that makes a stable map impossible.
In fact, on astronomical time scales the dynamics of the solar system (or generally, any multi-body gravitational system) displays such behaviors.