Introduction We start with some general questions to which we believe it is helpful for students to be able to formulate some kind of answers. The question for teachers of mathematics at all levels is to what extent, if at all, the training of mathematicians should involve professional discussion of, and assessment in, possible answers to these questions, such as those given or suggested here. Some basic issues for mathematicians Is mathematics important? If so, for what, in what contexts,and why? What is the nature of mathematics, in comparison with other subjects? What are the objects of study of mathematics? What is the importance of mathematics? It is not generally recognised how much of a part mathematics plays in our daily lives. Some of the mathematics is of course quite old: every day we use numbers, graphs, addition and multiplication. It is easy to forget that the invention of these was at one time a great discovery. The replacement of Ro...