I would rather be a novelist or a painter than a statesman of
similar rank; and there are many roads to fame which most of us would
reject as actively pernicious. Yet it is seldom that such differences of
value will turn the scale in man's choice of a career, which will almost
always be dictated by the limitations of his natural abilities. Poetry is
more valuable than cricket, but Bradman would be a fool if he sacrificed
his cricket in order to write second-rate minor poetry (and I suppose that
it is unlikely that he could do better). If the cricket were a little less
supreme, and the poetry better, then the choice might be more difficult: I
do not know whether I would rather have been Victor Trumper or Rupert
Brooke. It is fortunate that such dilemmas occur so seldom.
I may add that they are particularly unlikely to present themselves
to a mathematician. It is usual to exaggerate rather grossly the
differences between the mental processes of mathematicians and other
people, but it is undeniable that a gift for mathematics is one of the
most specialized talents, and that mathematicians as a class are not
particularly distinguished for general ability or versatility. If a man is
in any sense a real mathematician, then it is a hundred to one that his
mathematics will be far better than anything else he can do, and that he
would be silly if he surrendered any decent opportunity of exercising his
one talent in order to do undistinguished work in other fields. Such a
sacrifice could be justified only by economic necessity or age.
* * *
There are many highly respectable motives which may lead men to
prosecute research, but three which are much more important than the rest.
The first (without which the rest must come to nothing) is intellectual
curiosity, desire to know the truth. Then, professional pride, anxiety to
be satisfied with one's performance, the shame that overcomes any
self-respecting craftsman when his work is unworthy of his talent.
Finally, ambition, desire for reputation, and the position, even the power
of money, which it brings. It may be fine to feel, when you have done your
work, that you have added to the happiness or alleviated the sufferings of
others, but that will not be why you did it. So if a mathematician, or a
chemist, or even a physiologist, were to tell me that the driving force in
his work had been the desire to benefit humanity, then I should not
believe him (nor should I think the better of him if I did). His dominant
motives have been those which I have stated, and in which, surely, there
is nothing of which any decent man need be ashamed.
* * *
Archimedes will be remembered when Aeschylus is forgotten, because
languages die and mathematical ideas do not. 'Immortality' may be a silly
word, but probably a mathematician has the best chance of whatever it may
mean.
* * *
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.
* * *
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 and aesthetic appreciation of
mathematics is `a monomania confined to a few eccentrics in each
generation'.
It would be 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 maybe 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 may 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 note entirely, since psychology also plays a part), and everyone who
calls a problem `beautiful' is applauding mathematical beauty, even if it
is 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 further, 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.
* * *
The seriousness of a theorem, of course, does not lie in its
consequences, which are merely the evidence for its seriousness.
Shakespeare had an enormous influence on the development of the English
language, Otway next to none, but that is not why Shakespeare was the
better poet. He was the better poet because he wrote much better poetry.
The inferiority of the chess problem, like that of Otway's poetry, lies
not in its consequences but its content.
* * *
The proof is by reductio ad absurdum, and reductio ad absurdum,
which Euclid loved so much, is one of a mathematician's finest weapons. It
is a far finer gambit than any chess gambit: a chess player may offer the
sacrifice of a pawn or even a piece, but a mathematician offers the game.
* * *
Some measure of generality must be present in any high-class
theorem, but too much tends inevitably to insipidity. `Everything is what
it is, and not another thing', and the differences between things are
quite as interesting as their resemblances. We do not choose our friends
because they embody all the pleasant qualities of humanity, but because
they are the people that they are. And so in mathematics; a property
common to too many objects can hardly be very exciting, and mathematical
ideas also become dim unless they have plenty of individuality. Here at
any rate I can quote Whitehead on my side: `it is the large
generalization, limited by a happy particularity, which is the fruitful
conception.'
* * *
What `purely aesthetic' qualities can we distinguish in such
theorems as Euclid's and Pythagoras's? I will not risk more than a few
disjointed remarks.
In both theorems (and in theorems, of course, I include the proofs)
there is a very high degree of unexpectedness, combined with inevitability
and economy. The arguments take so odd and surprising a form; the weapons
used seem so childishly simple when compared with the far-reaching
results; but there is no escape from their conclusions. There are no
complications of detail - one line of attack is enough in each case; and
this is true too of the proofs of many much more diffcult theorems, the
full appreciation of which demands quite a high degree of technical
proficiency. We do not want many `variations' in the proof of a
mathematical theorem: `enumeration of cases', indeed, is one of the duller
forms of mathematical argument. A mathematical proof should resemble a
simple and clear-cut constellation, not a scattered cluster in the Milky
Way.
* * *
I believe that mathematical reality lies outside us, that our
function is to discover or observe it, and that the theorems which we
prove, and which we describe grandiloquently as our "creations," are
simply the notes of our observations.
* * *
Let us suppose that I am giving a lecture on some system of
geometry, such as the ordinary Euclidean geometry, and that I draw figures
on the blackboard to stimulate the imagination of my audience, rough
drawings of straight lines or circles or ellipses. It is plain, first,
that the truth of the theorems which I prove is in no way affected by the
quality of my drawings. Their function is merely to bring home my meaning
to my hearers, and, if I can do that, there would be no gain in having
them redrawn by the most skilful draughtsman. They are pedagogical
illustrations, not part of the real subject-matter of the lecture.
Now let us go a stage further. The room in which I am lecturing is a
part of the physical world, and has itself a certain pattern. The study of
that pattern, and of the general pattern of physical reality, is a science
in itself, which we may call `physical geometry'. Suppose now that a
violent dynamo, or a massive gravitating body, is introduced into the
room. Then the physicists tell us that the geometry of the room is
changed, its whole physical pattern slightly but definitely distorted. Do
the theorems which I have proved become false? Surely it would be nonsense
to suppose that the proofs of them which I have given are affected in any
way. It would be like supposing that a play of Shakespeare is changed when
a reader spills histea over a page. The play is independent of the pages
on which it is printed, and `pure geometries' are independent of lecture
rooms, or of any other detail of the physical world.
This is the point of view of a pure mathematician. Applied
mathematicians, mathematical physicists, naturally take a different view,
since they are preoccupied with the physical world itself, which also has
its structure or pattern. We cannot describe this pattern exactly, as we
can say that of a pure geometry, but we can say something significant
about it. We can describe, sometimes fairly accurately, sometimes very
roughly, the relations which hold between some of its constituents, and
compare them with the exact relations holding between constituents of some
system of pure geometry. We may be able to trace a certain resmeblance
between the two sets of relations, and then the pure geometry will become
interesting to physicists; it will give us, to that extent, a map which
`fits the facts' of the physical world. The geometer offers to the
physicist a whole set of maps from which to choose. One map, perhaps, will
fit the facts better than others, and then the geometry that provides that
particular map will be the geometry most important for applied
mathematics. I may add that even a pure mathematician may find his
appreciation of this geometry quickened, since there is no mathematician
so pure that he feels no interest at all in the physical world; but, in so
far as he succumbs to this temptation, he will be abandoning his purely
mathematical position.
* * *
It may be that modern physics fits best into some framework of
idealistic philosophy -- I do not believe it, but there are eminent
physicists who say so. Pure mathematics, on the other hand, seems to me a
rock on all which all idealism founders: 317 is a prime, not because we
think so, or because our minds are shaped in one way rather than another,
but because it is so, because mathematical reality is built that way.
* * *
One rather curious conclusion emerges, that pure mathematics is on
the whole distinctly more useful than applied. A pure mathematician seems
to have the advantage on the practical as well as on the aesthetic side.
For what is useful above all is technique, and mathematical technique is
taught mainly through pure mathematics.
* * *
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 do 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. I
have just one chance of escaping a verdict of complete triviality, that I
may be judged to have created something worth creating. And that I have
created something is undeniable: the question is about its value.
The case for my life, then, or for that of any one else who has been
a mathematician in the same sense in which I have been one, is this: that
I have added something to knowledge, and helped others to add more; and
that these somethings have a value which differs in degree only, and not
in kind, from that of the creations of the great mathematicians, or of any
of the other artists, great or small, who have left some kind of material
behind them.
* * *