There seem to be some very good reasons for a philosopher of science to be a deductivist about scientific reasoning. Deductivism is apparently connected with a demand for clarity and definiteness in the reconstruction of scientists' reasonings. And some philosophers even think that deductivism is the way around the problem of induction. But the deductivist image is challenged by cases of actual scientific reasoning, in which hard-to-state and thus discursively ill-defined elements of thought nonetheless significantly condition what practitioners accept as (...) cogent argument. And arguably, these problem cases abound. For example, even geometry--for most of its history--was such a problem case, despite its exactness and rigor. It took a tremendous effort on the part of Hilbert and others, to make geometry fit the deductivist image. Looking to the empirical sciences, the problems seem worse. Even the most exact and rigorous of empirical sciences--mechanics--is still the kind of problem case which geometry once was. In order for the deductivist image to fit mechanics, Hilbert's sixth problem (for mechanics) would need to be solved. This is a difficult, and perhaps ultimately impossible task, in which the success so far achieved is very limited. I shall explore some consequences of this for realism as well as for deductivism. Through discussing links between non-monotonicity, skills, meaning, globality in cognition, models, scientific understanding, and the ideal of rational unification, I argue that deductivists can defend their image of scientific reasoning only by trivializing it, and that for the adequate illumination of science, insights from anti-deductivism are needed as much as those which come from deductivism. (shrink)