Foreword

This is a book of philosophical thoughts about proofs, applications, and other mathematical activities.

Philosophers tend to emphasize mathematical ‘knowledge’, but as G. H. Hardy said on the first page of his Apology (1940), ‘the function of a mathematician is to do something, to prove new theorems, to add to mathematics’. I have emphasized the ‘do’. Hardy was writing not only an Apologia pro vita sua, but also a mathematician’s Lament that he was now too old to create much more mathematics. He also, notoriously, wanted to keep mathematics pure, whereas I believe that the uses, ‘the applications’, are as important as the theorems proved. Neither proof nor application is, however, as clear and distinct an idea as might be hoped.

To reflect on the doing of mathematics, on mathematics as activity, is not to practise the sociology of mathematics. Happily that is now a burgeoning field, from which one can learn much, but what follows is philosophizing, moved by old-fashioned questions – to which I add my title question, why do these questions arise perennially, from Plato to the present day?