This article outlines a philosophy of science in practice that focuses on the engineering sciences. A methodological issue is that these practices seem to be divided by two different styles of scientific reasoning, namely, causal-mechanistic and mathematical reasoning. These styles
are philosophically characterized by what Kuhn called ‘disciplinary matrices’. Due to distinct metaphysical background pictures and/or distinct ideas of what counts as intelligible, they entail distinct ideas of the character of phenomena and what counts as a scientific explanation.
It is argued that the two styles cannot be reduced to each other. At the same time, although they are incompatible, they must not be regarded as competing. Instead, they produce different kinds of epistemic results, which serve different kinds of epistemic functions. Moreover, some scientific
breakthroughs essentially result from relating them. This view of complementary styles of scientific reasoning is supported by pluralism about metaphysical background pictures.