Remarks on Mathematical Courseware

Computers are increasingly present in mathematical research and education. In research, they have opened up new territories previously inaccessible and spawned new fields; in education, they are being hailed as a revolutionary teaching aid. These two trends have, for the most part, developed separately. There is, however, a common thread which can bring them together: each depends for its success on discovering ways in which computers can support mathematical intuition. We sketch the role of intuition in the process of doing mathematics by contrasting it to logic, its polar principle. Looking more closely, it is possible to distinguish four freedoms which, when respected, allow a software tool to support mathematical intuition, whether its use be research or education. These are the freedoms.