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By G. H. Hardy

G. H. Hardy was once one in all this century's best mathematical thinkers, popular 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 an excellent and interesting account of arithmetic as greatly 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 wish to be an artistic artist'. C. P. Snow's Foreword provides sympathetic and witty insights into Hardy's lifestyles, with its wealthy shop of anecdotes referring to his collaboration with the intense Indian mathematician Ramanujan, his idiosyncrasies, and his ardour for cricket. this can be a special account of the fascination of arithmetic and of 1 of its such a lot compelling exponents nowa days.

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We can answer them, with rather surprising accuracy, but only by boring much deeper, leaving the integers above us for a while, and using the most powerful weapons of the modern theory of functions. Thus the theorem which answers our questions (the so-called ‘Prime Number Theorem’) is a much deeper theorem than Euclid’s or even Pythagoras’s. I could multiply examples, but this notion of ‘depth’ is an elusive one even for a mathematician who can recognize it, and I 10 13 10 It is supposed that the number of protons in the universe is about 1080 .

It can be maintained that modern warfare is less 18 J. B. S. Haldane, Callinicus: a Defence of Chemical Warfare (1924). 44 horrible than the warfare of pre-scientific times; that bombs are probably more merciful than bayonets; that lachrymatory gas and mustard gas are perhaps the most humane weapons yet devised by military science; and that the orthodox view rests solely on loos-thinking sentimentalism19. It may also by urged (though this was not one of Haldane’s theses) that the equalization of risks which science was expected to bring would be in the long range salutary; that a civilian’s life is not worth more than a soldier’s, nor a woman’s more than a man’s; that anything is better than the concentration of savagery on one particular class; and that, in short, the sooner war comes ‘all out’ the better.

17 The second quality which I demanded in a significant idea was depth, and this is still more difficult to define. It has something to do with difficulty; the ‘deeper’ ideas are usually the harder to grasp: but it is not at all the same. The ideas underlying Pythagoras’s theorem and its generalization are quite deep, but no mathematicians now would find them difficult. e. theorems about the solution of equations in integers). It seems that mathematical ideas are arranged somehow in strata, the ideas in each stratum being linked by a complex of relations both among themselves and with those above and below.

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