For arriving at a better understanding of the Peircean notion of diagrammatic reasoning there appear to be two complementary ways. One way is to substantiate its impact and relevance by interpreting actual mathematical reasoning as being diagrammatic. This in fact can be done in a great variety of cases. Another way is to exhibit cases of mathematical notions, concepts and arguments which inherently do not lend themselves in a direct way to diagrammatic reasoning. Analyzing those examples will again sharpen and refine the notions of diagram and diagrammatic reasoning. Or, it might possibly point to the necessity of widening those notions to comprise also the manipulation of words and linguistic terms according to specific rules. The latter two topics are the main issues treated in the paper.