The Mind of the Mathematician

Chapter 1. Mathematicians and Their World

3 Chapter 1 } Mathematicians and Their World The Attraction of Mathematics A mathematician, according to the Oxford English Dictionary, is someone who is skilled in mathematics. Such a person may be involved in teaching or research, or in applications of the discipline. He or she may use mathematics to make a living, or enjoy it more as a form of recreation. Until relatively recently, mathematics embraced much of what we now call natural science, and most mathematicians were actively interested in its applications even if they mainly worked on the pure side of the discipline. Some, however, regarded mathematics as an end in itself, and there is a tendency in the literature to write as if that were the general belief. In this book, we use the term ‘‘mathematics’’ in the broad sense, to include both pure and applied mathematics, and to some extent we also include mathematical physics, statistics, and computer science. It is di≈cult to estimate the number of mathematicians in the world today, but there must be well over a million. Only a minority, perhaps fifty thousand, could be described as active in research, and of these only a much smaller minority, perhaps five thousand, are publishing research of lasting value. Some of them—not very many—are truly creative mathematicians, who become famous primarily because of their excellence in research. It is these creative mathematicians who loom so large in the history of mathematics , as they will in this book, but we must not forget that they form only a small, atypical minority of the many who describe themselves as mathematicians . We must be careful not to make sweeping statements about all mathematicians on the basis of what we know about this minority. We must 4 Tour of the Literature also be careful not to make generalizations on the basis of what we know about pure mathematicians, especially that select group of specialists in the theory of numbers. Currently some eighty or ninety thousand research papers in the mathematical sciences are published every year, but the majority of these are not in pure mathematics. Ordinary mathematicians do not care much about philosophical questions ; they leave that to the mathematical philosophers. However, if challenged , they might at least be willing to say whether they are Platonists or not. Platonists believe that in addition to objects, there exists a world of concepts to which we have access by intuition. For mathematical Platonists, numbers exist independently of ourselves in some objective sense. The Platonist mathematician tries to discover properties that numbers already have. The non-Platonist regards numbers as entirely a construction of the human mind, endowed with properties that can be investigated, so that the aim of research is to create or invent mathematics rather than discover it. The debate between the two points of view goes back to classical times. The philosophical literature defending and attacking Platonism in mathematics is too vast for us to pursue the matter further here, but the reader may wish to consult the discussion about the types of reality of mathematical entities in Changeux and Connes (1995). As mathematics continues to develop, it becomes increasingly di≈cult to capture its nature in a single definition, but people keep trying. The British mathematician G. H. Hardy described mathematicians as makers of patterns of ideas. A similar point of view was expressed by the mathematical philosopher A. N. Whitehead (1911) when he wrote: ‘‘The notion of the importance of patterns is as old as civilisation . . . mathematics is the most powerful technique for the understanding of patterns, and for the analysis of the relation of patterns.’’ Mathematics, he maintained, defines and gives names to these patterns, which generally originate in the physical world, so that we can manipulate them in our own minds and communicate ideas about them more easily. Other ideas about the nature of mathematics are discussed in the literature, but none of them seem entirely satisfactory. Some of them fail to take into account the role of the mathematician in the establishment of mathematical knowledge. Nor do they allow for the development of our knowledge of mathematics over time. Again, the reader may wish to consult the book by Changeux and Connes (1995). Mathematics, like other sciences, advances by correcting and re-correct- Mathematicians and Their World 5 ing mistakes, but as the French mathematician Elie Cartan (1952–1955) explained, ‘‘A desire to avoid mistakes forces mathematicians...