[eriktaubeneck]

That's an interesting point I had not considered. My view (mostly from the math side, my shallow dive into physical systems was just from a PDE class in grad school) is from the view of unstable differential equations where very small changes to initial conditions can cause massive changes to the output of the system. My uninformed instincts would tell me that even with orders of more digits than atoms in a physical system, changes smaller than the available accuracy to initial conditions could have significant impact on the eventual state of the system.

[replicant]

Another point of view on why it is not striking that we can model heat conduction well. The heat equation is well-posed, which among other things, means that the solution depends continuously of the input data. And we observe this in many physical systems, that small changes in the cause produces small changes in the effect, therefore, we should expect from good mathematical models of reality to display this property.

Now, orbital mechanics do display unstable behaviour. I don't dare to adventure on how people work around this. https://en.wikipedia.org/wiki/Well-posed_problem