By G. H. Hardy

G. H. Hardy used to be one among this century's most interesting mathematical thinkers, well known between his contemporaries as a 'real mathematician ... the purest of the pure'. He was once additionally, as C. P. Snow recounts in his Foreword, 'unorthodox, eccentric, radical, able to speak about anything'. This 'apology', written in 1940, bargains a super and interesting account of arithmetic as greatly greater than a technology; while it used to be first released, Graham Greene hailed it along Henry James's notebooks as 'the most sensible account of what it used to be wish to be an inventive artist'. C. P. Snow's Foreword provides sympathetic and witty insights into Hardy's existence, with its wealthy shop of anecdotes relating his collaboration with the bright Indian mathematician Ramanujan, his idiosyncrasies, and his ardour for cricket. it is a precise account of the fascination of arithmetic and of 1 of its so much compelling exponents nowa days.

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The trivial mathematics may be justified by arguments which would appeal to Hogben, or other writers of his school, but there is no such defence for the real mathematics, which must be justified as arts if it can be justified at all. There is nothing in the least paradoxical or unusual in this view, which is that held commonly by mathematicians. We have still one more question to consider. We have concluded that the trivial mathematics is, on the whole, useful, and that the real mathematics, on the whole, is not; that the trivial mathematics does, and the real mathematics does not, ‘do good’ in a certain sense; but we have still to ask whether either sort of mathematics does harm.

This is a question which can be answered definitely and about which there is general agreement among mathematicians. There will be nothing in the least unorthodox about my answer, but it needs a little preface. My next two sections will have a mildly philosophical flavour. The philosophy will not cut deep, or be in any way vital to my main theses; but I shall use words which are used very frequently with definite philosophical implications, and a reader might well become confused if I did not explain how I shall use them.

20 All this may seem very obvious, but even here there is often a good deal of confusion, since the most ‘useful’ subjects are quite commonly just those which it is most useless for most of us to learn. It is useful to have an adequate supply of physiologists and engineers; but physiology and engineering are not useful studies for ordinary men (though their study may of course be defended on other grounds). For my own part I have never once found myself in a position where such scientific knowledge as I possess, outside pure mathematics, has brought me the slightest advantage.

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