ABSTRACT
Mathematics has many dierent aspects, properties, and features, so it is probably impossible to answer succinctly the question `what is mathematics?' There are, however, properties of the subject that spring readily to mind: most would agree that mathematics is abstract, rigorous, precise, and formal. No doubt, many would also add `dicult' to that list. Mathematics is abstract because it deals with concepts rather than `real' objects. For example, mathematics concerns itself with the number 3 rather than three apples or three buttons; it deals with `idealised' triangles rather than imperfectly drawn `real' ones, and so on. It is rigorous and precise because it does not accept vague or emotional arguments; rather, `facts' about mathematical objects are only established by logical reasoning. The sum of angles in a Euclidean triangle is 180, not because that would be a nice thing to have or because we want it to be so; we know the angle sum is 180 because we can prove this to be the case from the basic assumptions of Euclidean geometry. Mathematics is formal because mathematical objects are given explicit and precise denitions and mathematical reasoning has a denite and systematic structure.
