The role of mental argumentation in Mathematics vis-à-vis property perception and the operational mode

In this article, mental argumentation concerning the doing of mathematics is characterized by that part of reasoning dealing with the perception of properties, in contrast to that involving an operational mode of thinking. In particular, the role of mental argumentation is argued in terms of semantics, holism, the usage of symbolism and what reasoning can be achieved without pen and paper. The discourse is based on a succession of mathematical tasks, stated in the text together with solutions and comments on how the solutions could be reached. What transpires is that there is a mutual dependence between mental argumentation and operational working, even though at first the two might seem to represent opposites. A partial account of the nature of this interaction is given.