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By G. H. Hardy

G. H. Hardy was once one among this century's most interesting mathematical thinkers, well known between his contemporaries as a 'real mathematician ... the purest of the pure'. He used to be additionally, as C. P. Snow recounts in his Foreword, 'unorthodox, eccentric, radical, able to discuss anything'. This 'apology', written in 1940, bargains an excellent and fascinating account of arithmetic as a great deal greater than a technological know-how; while it was once first released, Graham Greene hailed it along Henry James's notebooks as 'the top account of what it used to be prefer to be an artistic artist'. C. P. Snow's Foreword supplies sympathetic and witty insights into Hardy's existence, with its wealthy shop of anecdotes touching on his collaboration with the bright Indian mathematician Ramanujan, his idiosyncrasies, and his ardour for cricket. this can be a exact account of the fascination of arithmetic and of 1 of its so much compelling exponents nowa days.

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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. I have often used the adjective ‘real’, and as we use it commonly in conversation. I have spoken of ‘real mathematics’ and ‘real mathematicians’, as I might have spoken of ‘real poetry’ or ‘real poets’, and I shall continue to do so.

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. I have often used the adjective ‘real’, and as we use it commonly in conversation. I have spoken of ‘real mathematics’ and ‘real mathematicians’, as I might have spoken of ‘real poetry’ or ‘real poets’, and I shall continue to do so. But I shall also use the word ‘reality’, and with two different connotations.

But our general conclusion must be that such mathematics is useful as is wanted by a superior engineer or a moderate physicist; and that is roughly the same thing as to say, such mathematics as has no particular aesthetic merit. Euclidean geometry, for example, is useful in so far as it is dull—we do not want the axiomatics of parallels, or the theory of proportion, or the construction of the regular pentagon. 40 One rather curious conclusion emerges, that pure mathematics is one the whole distinctly more useful than applied.

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