Our cognitive capabilities force us into a description of the world by levels. But theories on different levels result in descriptions that differ qualitatively. Therefore, the resulting incommensurability requires ontological bridges between such theories. These are obtained uniquely when the equations of the reduced theory are compared with a suitable limit of the equations of the reducing theory. Four case studies from theoretical physics and astronomy support this claim, two for theories of composites and two for non-composites (field theories). These (...) results a coherent view of a single real world despite its ontological pluralism. The cumulativity of scientific knowledge is thus ensured and realism is supported. (shrink)