ABSTRACT
The question of what someone is doing in carrying out a mathematical inquiry is one of those simple-looking questions the answer to which can rapidly become rather complicated both philosophically and procedurally. Philosophically, it depends on one’s conception of what mathematics is, even down to the language used in describing mathematical inquiry: to speak of discovering mathematics generally implies a belief that mathematical truths exist independently of our minds, while inventing mathematics carries the suggestion that it is a human construction. This problem is not unique to mathematics-photographers have been arguing for a century about whether a photograph is taken or made-but an awareness of the issues is particularly important in understanding the place of mathematics in other kinds of inquiry. The aim of this chapter is to provide a number of contrasting views of mathematical activity, in which it may be seen how each is related to a particular conception not only of what mathematics is but also of what value or purpose it has.
