By G. H. Hardy

*A Mathematician's Apology* is the recognized essay by way of British mathematician G. H. Hardy. It matters 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 e-book is usually certainly one of the easiest insights into the brain of a operating mathematician written for the layman.

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It seems to me again that there is only one possible answer: yes, perhaps, but, if so, for one reason only: I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world. I have helped to train other mathematicians, but mathematicians of the same kind as myself, and their work has been, so far at any rate as I have helped them to it, as useless as my own. Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow.

19 I must return to my Oxford apology, and examine a little more carefully some of the points which I postponed in §6. It will be obvious by now that I am interested in mathematics only as a creative art. But there are other questions to be considered, and in particular that of the ‘utility’ (or uselessness) of mathematics, about which there is much confusion of thought. We must also consider whether mathematics is really quite so ‘harmless’ as I took for granted in my Oxford lecture. A science or an art may be said to be ‘useful’ if its development increases, even indirectly, the material well-being and 15 I believe that it is now regarded as a merit in a problem that there should be many variations of the same type.

Thus the idea of an ‘irrational’ is deeper than that of an integer; and Pythagoras’s theorem is, for that reason, deeper than Euclid’s. Let us concentrate our attention on the relations between the integers, or some other group of objects lying in some particular stratum. Then it may happen that one of these relations can be comprehended completely, that we can recognize and prove, for example, some property of the integers, without any knowledge of the contents of lower strata. Thus we proved Euclid’s theorem by consideration of properties of integers only.