G.H. Hardy

Godfrey Harold Hardy attended Cranleigh, followed by Trinity College, Cambridge, becoming a fellow of the College the following year and continued to lecture there until 1919. Throughout his academic life he never tired of what he saw as the beauty of mathematics. He developed theories and analysed his subject in a fresh and new way, changing the way the world looked at British mathematicians, who were known as dwellers in applied mathematics before his time. His many essays, the most famous of which is A Mathematician’s Apology, also brought changes in lay understanding of mathematics.

A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs it is because they are made with ideas. A painter makes patterns with shapes and colours, a poet with words. A painting may embody an ‘idea’, but the idea is usually commonplace and unimportant. In poetry, ideas count for a good deal more; but, as Housman insisted, the importance of ideas in poetry is habitually exaggerated: ‘I cannot satisfy myself that there are any such things as poetical ideas… Poetry is not the thing said but a way of saying it.’

Not all the water in the rough rude sea
Can wash the balm from an anointed king

Could the line be better, and could ideas be at once more trite and more false? The poverty of the ideas seems hardly to affect the beauty of the verbal pattern. A mathematician, on the other hand, has no material to work with but ideas, and so his patterns are likely to last longer, since ideas waste less time than words.

The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics. And here I must deal with a misconception which is still widespread (though probably much less so now than it was twenty years ago), what Whitehead has called the ‘literary superstition’ that love of an aesthetic appreciation of mathematics is a ‘monomania confined to a few eccentrics in each generation.’ It would be quite difficult now to find an educated man quite insensitive to the aesthetic appeal of mathematics. It may be very hard to define mathematical beauty, but that is just as true of beauty of any kind – we may not know quite what we mean by a beautiful poem, but that does not prevent us from recognising one when we read it. Even Professor Hogben, who is out to minimise at all costs the importance of the aesthetic element in mathematics, does not venture to deny its reality. ‘There are, to be sure, individuals for whom mathematics exercises a coldly impersonal attraction…The aesthetic appeal of mathematics may be very real for a chosen few.’ But they are ‘few’, he suggests, and they feel ‘coldly’ (and are really rather ridiculous people, who live in silly little university towns sheltered from the fresh breezes of the wide open spaces). In this he is merely echoing Whitehead’s ‘literary superstition’.

The mathematician’s patterns, like the painter’s or the poet’s must be beautiful

The fact is that there are few more ‘popular’ subjects than mathematics. Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune; and there are probably more people interested in mathematics than in music. Appearances suggest the contrary, but there are easy explanations. Music can be used to stimulate mass emotion, while mathematics cannot; and musical incapacity is recognised (no doubt rightly) as mildly discreditable, whereas most people are so frightened by the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity.

A very little reflection is enough to expose the absurdity of the ‘literary superstition’. There are masses of chess players in every civilised country – in Russia, almost the whole educated population; and every chess player can recognise and appreciate a ‘beautiful’ game or problem. Yet a chess problem is simply an exercise in pure mathematics (a game not entirely, since psychology also plays a part), and everyone who calls a problem ‘beautiful’ is applauding mathematical beauty, even if it is a beauty of a comparatively lowly kind. Chess problems are the hymn-tunes of mathematics.

We may learn the same lesson, at a lower level but for a wider public, from bridge, or descending farther, from the puzzle columns of the popular newspapers. Nearly all their immense popularity is a tribute to the drawing power of rudimentary mathematics, and the better makers of puzzles, such as Dudeney or ‘Caliban’, uses very little else. They know their business: what the public wants is a little intellectual ‘kick’, and nothing else has quite the kick of mathematics.