Well, if you formulate the statement as: > We do know that Newtonian mechanics is not deterministic at singularity points then I fully agree with that. Howeve

Well, if you formulate the statement as: > We do know that Newtonian mechanics is not deterministic at singularity points then I fully agree with that. However, that may be fixed by either adding additional requirement (e.g. Lipschitz continuity) or just by not considering Newtonian mechanics applicable to those cases - it is known already that Newton's laws do not fully describe the real word (because quantum uncertainty does exist) and the Lebesgue measure of singularity cases is zero anyway. In case of three-body problem, the singularities are the case of bodies collisions and yes, those cases are not deterministic, but the configuration without collision is known to be fully deterministic.