A Mathematician’s Apology by GHHardy

Chapter 10

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 and ‘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 no 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 lines 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 wear less with 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 recognizing one
when we read it. Even Professor Hogben, who is out to minimize
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 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 really 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 recognized (no
doubt rightly) as mildly discreditable, whereas most people are so
frightened of 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
civilized country—in Russia, almost the whole educated
population; and every chess-player can recognize 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’, use 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.

I might add that there is nothing in the world which pleases
even famous men (and men who have used quite disparaging
words about mathematics) quite so much as to discover, or
rediscover, a genuine mathematical theorem. Herbert Spencer
republished in his autobiography a theorem about circles which
he proved when he was twenty (not knowing that it had been
proved over two thousand years before by Plato). Professor
Soddy is a more recent and more striking example (but his
theorem really is his own).