By G. H. Hardy

*A Mathematician's Apology* is the recognized essay by way of British mathematician G. H. Hardy. It issues the aesthetics of arithmetic with a few own content material, and offers the layman an perception into the brain of a operating mathematician. certainly, this publication is frequently one of the easiest insights into the brain of a operating mathematician written for the layman.

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33 22 There is another misconception against which we must guard. It is quite natural to suppose that there is a great difference in utility between ‘pure’ and ‘applied’ mathematics. This is a delusion: there is a sharp disctinction between the two kinds of mathematics, which I will explain in a moment, but it hardly affects their utility. How do pure and applied mathematicians differ from one another? This is a question which can be answered definitely and about which there is general agreement among mathematicians.

It is true that there are branches of applied mathematics, such as ballistics and aerodynamics, which have been developed deliberately for war and demand a quite elaborate technique: it is perhaps hard to call them ‘trivial’, but none of them has any claim to rank as ‘real’. They are indeed repulsively ugly and intolerably dull; even Littlewood could not make ballistics respectable, and if he could not who can? So a real mathematician has his conscience clear; there is nothing to be set against any value his work may have; mathematics is, as I said at Oxford, a ‘harmless and innocent’ occupation.

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.