The Methodology of Mathematics

by Ronald Brown and Timothy Porter

Introduction

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 to 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 of these issues are discussed in the books by Davis and Hersh [2,3].

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 methodology of mathematics, what is the way it goes about its job? Is there research going on in mathematics? If so, how much? What are its broad aims or main aims? What are its most important achievements? How does one go about doing mathematical research? What is good mathematics?

It may be thought by some that these questions are beside the point, a waste of time, and not what a real mathematician should be considering. Against this we would like to give a quotation from Albert Einstein (1916) [4]: