This essay is based on a talk given by the first author to
students and staff of the Departmento de Geometria e Topologia
at the University of Seville in November, 1993. The issues
presented there have been part of a continued debate and
discussion at Bangor over many years, and this explains why this
is a joint paper. The aim of the talk, and the reason for
discussing these topics, was to give students an understanding
and a sense of pride in the aims and achievements of their
subject, and so help them explain these aims and achievements to
their friends and relatives. This pride in itself would be
expected to contribute to their enjoyment of the subject,
whatever their own level of achievement. Because of this, and
because of its origin, the tone of the article is principally
that of an address to students. We do not claim to be alone in
addressing these questions. Dr Allan Muir (City University) has
organised a ``How Mathematics works'' group for some years, and
there is a similar group in the U.S.A. Many