[magicalist]

> but the approximate present does not approximately determine the future

Again, though, that statement contains huge assumptions about the nature of the dynamical system and how approximate a future prediction we need to call it "accurately predicted". Errors accumulate, but we can get arbitrarily close by more accurately measuring where we are in phase space (which is what the OP was talking about) and by more accurately modeling the system. There will always be errors -- dynamics are hard -- but we can certainly reduce them for some distance into the future. That's why we can successfully put things into orbit around Mars, for instance, in spite of the many interacting bodies in our solar system.

Assuming an increase in accuracy even somewhat proportional to the increase in sensors will almost definitely turn out to be wrong, though, which may be more the point you're trying to make and I'm missing.