With two interacting bodies, interacting via the r-squared forces of gravity or electricity, and obeying classical mechanics, the equation of motion can be written down in closed form.

With three interacting bodies, the equation of motion can’t be written down in closed form.

So the three-body problem can’t be solved exactly (and this is just for classical mechanics and r-squared forces). Algebraic approximations or numerical simulations can do a good job, but they’re not exact.

So what’s it mean?

Our knowledge in principle: the three-body problem is not reductionistic. It can’t be perfectly described by reducing it to its parts. But is it holistic – are there holistic rules that govern things? Or, how much can we say about the motions using holistic principles? Just Cons E, statistical measures like entropy, and stuff like that?

Our knowledge in practice: the three-body problem can be treated reductionisticly.

How the universe does it: unknown if it’s holistic or reductionistic.

What about many-body problems? Atoms are a nice example.

Our knowledge in principle: like the three-body problem, it’s not reductionistic in principle, and we have to resort to holistic things in principle. Also there’s quantum mechanics and relativity, which change things (how?).

Our knowledge in practice: it’s easier to use the periodic table. Though in many cases we need to get reductionistic as a supplement or even skip the periodic table almost entirely and use reductionistic methods, even though they’re not exact/in principle.

How the universe does it: unknown if it’s holistic or reductionistic.

So this example is purely at “one level” of science: physics, or at two levels: physics and chemistry.