His statement that (freely paraphrasing) "seeing everything, and having the intellect to analyze the data, allows for knowledge of the future" brings up an interesting difficulty that I recalled today, while reviewing an orbital mechanics text.
The immediate context was that, while the motion of a body subject to central body gravitation is completely solveable, an "imposed non-two-body acceleration ... will render the new system [of equations] insolvable." (Emphasis mine.)
The author's essential point is that imposing perturbations other than gravity leaves us with a trajectory problem having more unknowns than parameters to explain the motion in a closed-form way. (This explains why there's no solution to the n-body problem, for example.)
It's an interesting lesson on the limitations of mathematics as they apply to the real world. At best, Laplace's statement boils down to a statement of perfect measurement of an immense number of initial conditions; coupled with zero-error numerical prediction methods -- neither of which are attainable in the real world.
Indeed r9etb -- thank you so much for this telling insight!