Godfrey Harold Hardy’s memoir, *A Mathematician’s Apology*, was published when he was 63 years old. It is a slight volume at just 90 pages, but is replete with interesting observations and not a few controversial opinions. After 78 years, it is still in print and is available in virtually every mathematics library. Though many of Hardy’s opinions are difficult to support and some of his predictions have turned out to be utterly wrong, the book is still well worth reading.

**Over the Hill**

Hardy (1877-1947) was one of the great mathematicians of the early twentieth century and a leading light in British mathematics. His book was written after his most creative period had passed and he is clearly very saddened by his fading mathematical powers. He admits that he no longer has the freshness of mind or energy that were so evident in his earlier years. He is clearly unhappy and uneasy to be writing about mathematics rather than doing mathematics. Indeed, he notes the scorn of those who create mathematics for those who merely explain it.

For Hardy, expository writing on mathematics or anything else is work for second-rate minds (that puts me in my box). Hardy supports his view that mathematics “is a young man’s game” by giving several examples. However, he overstates the case: there are now many examples of significant advances by mathematicians aged far beyond their forties.

**Aesthetics**

In * A Mathematician’s Apology*, Hardy argued that beauty is the over-riding criterion for the value of a mathematical result. “A mathematician, like a painter or poet, is a maker of patterns. The patterns, like those of the painter or poet, must be beautiful. There is no permanent place in the world for ugly mathematics.” Hardy gave some criteria for the significance of mathematical proofs. He suggested that a beautiful proof has “inevitability, unexpectedness and economy”: it sparkles with surprise, with intriguing connections between disparate areas of mathematics.

Hardy was right to stress aesthetics as a dominant motivation and a primary criterion of value. But he was lamentably short-sighted in dismissing applicable mathematics as trivial. Beauty is quite subjective, and Hardy saw all practically useful mathematics as ugly: “Very little of mathematics is practically useful, and that little is comparatively dull.” This is not a view that would be shared by many today.

**Inapplicable Mathematics**

Hardy was well aware of the great practical value of mathematics – he mentions bridges and steam-engines and dynamos – but he repeatedly describes these applications as “crude achievements”. He claimed that useful mathematics was “repulsively ugly and intolerably dull”, with no aesthetic merit whatsoever.

Hardy was inclined to revel in the “uselessness” of his work, bragging about its lack of practical applications in ordinary life. He wrote “I have never done anything useful. No discovery of mine has made … the least difference to the amenity of the world.” It is probably this attitude that infuriated one reviewer of Hardy’s book, who wrote that “from such cloistral clowning the world sickens”.

Hardy argues that while a mathematician’s work may not be financially profitable, at least it does no harm. And the best mathematics has a quality of *permanence* not found in the other sciences: it is the oldest and the youngest study. Hardy cites two shining examples to illustrate the remarkable endurance of mathematical theorems: the infinitude of the prime numbers and the irrationality of the square root of 2. These results, and their ingenious proofs, are as fresh and as significant today as when they were first discovered.