By G. H. Hardy
G. H. Hardy used to be considered one of 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 discuss anything'. This 'apology', written in 1940, deals an excellent and fascinating account of arithmetic as greatly greater than a technology; 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 offers sympathetic and witty insights into Hardy's lifestyles, with its wealthy shop of anecdotes referring to his collaboration with the bright Indian mathematician Ramanujan, his idiosyncrasies, and his ardour for cricket. this can be a targeted account of the fascination of arithmetic and of 1 of its such a lot compelling exponents nowa days.
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But the lectures and demonstrations arranged for this purpose were rather a fiasco. It appeared that the members (whether citizens of Leeds or not) wanted to be entertained, and the ‘heavy wool’ is not at all an entertaining subject. So the attendance at these lectures was very disappointing; but those who lectured on the excavations at Knossos, or on relativity, or on the theory or prime numbers, were delighted by the audiences that they drew. 26 What parts of mathematics are useful? First, the bulk of school mathematics, arithmetic, elementary algebra, elementary Euclidean geometry, elementary differential and integral calculus.
It makes no difference to a chess problem whether the pieces are white and black, or red and green, or whether there are physical ‘pieces’ at all; it is the same problem which an expert carries easily in his head and which we have to reconstruct laboriously with the aid of the board. The board and the pieces are mere devices to stimulate our sluggish imaginations, and are no more essential to the problem than the blackboard and the chalk are to the theorems in a mathematical lecture. It is not this kind of generality, common to all mathematical theorems, which we are looking for now, but the more subtle and elusive kind of generality which I tried to describe in rough terms 26 in §15.
We are looking for differences of generality between one mathematical theorem and another, and in Whitehead’s sense all are equally general. Thus the ‘trivial’ theorems (a) and (b) of §15 are just as ‘abstract’ or ‘general’ as those of Euclid and Pythagoras, and so is a chess problem. It makes no difference to a chess problem whether the pieces are white and black, or red and green, or whether there are physical ‘pieces’ at all; it is the same problem which an expert carries easily in his head and which we have to reconstruct laboriously with the aid of the board.
A Mathematician’s Apology by G. H. Hardy