IT is time to begin thinking about the first question which I put in § 3, and which is so much more difficult than the second. Is mathematics, what I and other mathematicians mean by mathematics, worth doing; and if so, why?

I have been looking again at the first pages of the inaugural lecture which I gave at Oxford in 1920, where there is an outline of an apology for mathematics. It is very inadequate (less than a couple of pages), and it is written in a style (a first essay, I suppose, in what I then imagined to be the ‘Oxford manner’) of which I am not now particularly proud; but I still feel that, however much development it may need, it contains the essentials of the matter. I will resume what I said then, as a preface to a fuller discussion.

(1) I began by laying stress on the harmlessness of mathematics— ‘the study of mathematics is, if an unprofitable, a perfectly harmless and innocent occupation’. I shall stick to that, but obviously it will need a good deal of expansion and explanation.

Is mathematics ‘unprofitable’? In some ways, plainly, it is not; for example, it gives great pleasure to quite a large number of people. I was thinking of ‘profit’, however, in a narrower sense. Is mathematics ‘useful’, directly useful, as other sciences such as chemistry and physiology are?