ABSTRACT
One interesting aspect of the mathematical activity is the great economy with which the experts communicate their arguments to one another. It presupposes a vague but useful distinction between the new and the routine. Also in memorizing an argument, a good mathematician tends to retain just the barest outline which is suffic ient to recall, when the need arises, a whole long chain of details. This indicates a very flexible structure in the head which is almost never revealed in written works but which could sometimes be taught in a highly informal manner. A related ability is to use rich general concepts in a variety of diverse contexts; for example, an open set, a diagonal argument, a tree.
