A Mathematician’s Apology
by G. H. Hardy
I HAD better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man’s game. To take a simple illustration at a comparatively humble level, the average age of election to the Royal Society is lowest in mathematics.
We can naturally find much more striking illustrations. We may consider,
for example, the career of a man who was certainly one of the world’s three
greatest mathematicians. Newton gave up mathematics at fifty, and had lost
his enthusiasm long before; he had recognized no doubt by the time that he
was forty that his great creative days were over . His greatest ideas of
all, fluxions and the law of gravitation, came to him about 1666, when he
was twenty-four ---‘in those days I was in the prime of my age for invention
, and minded mathematics and philosophy more than at any time since’. He
made big discoveries until he was nearly forty ( the ‘elliptic orbit’ at
thirty-seven ), but after that he did little but polish and perfect .
Galois died at twenty-one , Abel at twenty-seven , Ramanujan at thirty-three , Riemann at forty . There have been men who have done great work a good deal later ; Gauss’s great memoir on differential geometry was published when he was fifty (though he had had the fundamental ideas ten years before). I do not know an instance of a major mathematical advance initiated by a man past fifty . If a man of mature age loses interest in and abandons mathematics , the loss is not likely to be very serious either for mathematics or for himself .
On the other hand the gain is no more likely to be substantial; the later records of mathematicians who have left mathematics are not particularly encouraging . Newton made a quite competent Master of the Mint ( when he was not quarrelling with anybody). Painleve was a not very successful Premier of France . Laplace’s political career was highly discreditable , but he is hardly a fair instance , since he was dishonest rather than incompetent, and never really ‘gave up’ mathematics . It is very hard to find an instance of a first-rate mathematician who has abandoned mathematics and attained first-rate distinction in any other field. There may have been young men who would have been first-rate mathematicians if they had stuck to mathematics , but I have never heard of a really plausible example. And all this is fully borne out by my own very limited experience . Every young mathematician of real talent whom I have known has been faithful to mathematics , and not from lack of ambition but from abundance of it ; they have all recognized that there , if anywhere , lay the road to a life of any distinction.
What we do may be small , but it has a certain character of permanence ; and to have produced anything of the slightest permanent interest , whether it be a copy of verses or a geometrical theorem , is to have done something utterly beyond the powers of the vast majority of men .
I shall assume that I am writing for readers who are full , or have in the past been full, of a proper spirit of ambition . A man’s first duty , a young man’s at any rate , is to be ambitious. Ambition is a noble passion which may legitimately take many forms; there was something noble in the ambition of Attila or Napoleon: but the noblest ambition is that of leaving behind one something of permanent value---
Ambition has been the driving force behind nearly all the best work of the world. In particular , practically all substantial contributions to human happiness have been made by ambitions men. To take two famous examples, were not Lister and Pasteur ambitions ? Or, on the humbler level , King Gillette and William Willett; and who in recent times have contributed more to human comfort than they ?
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 or the 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.
If intellectual curiosity , professional pride , and ambition are the dominant incentives to research , then assuredly no one has a fairer chance of gratifying them than a mathematician. His subject is the most curious of all --- there is none in which truth plays such odd pranks. It has the most elaborate and the most fascinating technique, and gives unrivalled openings for the display of sheer professional skill. Finally , as history proves abundantly , mathematical achievement, whatever its intrinsic worth , is the most enduring of all.
We can see this even in semi-historic civilizations . The Babylonian and Assyrian civilizations have perished ; Hammurabi, Sargon, and Nebuchadnezzar are empty names; yet Babylonian mathematics is still interesting, and the Babylonian scale of 60 is still used in astronomy. But of course the crucial case is that of the Creeks.
The Creeks were the first mathematicians who are still ‘real’ to us to-day . Oriental mathematics may be an interesting curiosity, but Creek mathematics is the real thing . The Greeks first spoke a language which modern mathematicians can understand; as Littlewood said to me once, they are not clever schoolboys or ‘scholarship candidates’, but ‘Follows of another college’. So Creek mathematics is ‘permanent’, more permanent even than Greek literature. 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.
Nor need he fear very seriously that the future will be unjust to him. Immortality is often ridiculous or cruel: few of us would have chosen to be Og or Ananias or Gallio . Even in mathematics , history sometimes plays strange tricks ; Rolle figures in the text-books of elementary calculus as if he had been a mathematician like Newton; Farey is immortal because he failed to understand a theorem which Haros had proved perfectly fourteen years before; the names of five worthy Norwegians still stand in Abel’s Life , just for one act of conscientious imbecility, dutifully performed at the expense of their country’s greatest man. But on the whole the history of science is fair, and this is particularly true in mathematics. No other subject has such clear-cut or unanimously accepted standards, and the men who are remembered are almost always the men who merit it. Mathematical fame, if you have the cash to pay it, is one of the soundest and steadiest of investments.
ALL this is very comforting for dons , and especially for professors of mathematics . It is sometimes suggested , by lawyers or politicians or business men, that an academic career is one sought mainly by cautious and unambitious persons who care primarily for comfort and security . The reproach is quite misplaced . A don surrenders something , and in particular the chance of making large sums of money---it is very hard for a professor to make ￡2000 a year; and security of tenure is naturally one of the considerations which make this particular surrender easy. That is not why Housman would have refused to be Lord Simon or Lord Beaverbrook. He would have rejected their careers because of his ambition , because he would have scorned to be a man to be forgotten in twenty years .
Yet how painful it is to feel that, with all these advantages , one may fail. I can remember Bertrand Russell telling me of a horrible dream. He was in the top floor of the University Library, about A.D. 2100. A library assistant was going round the shelves carrying an enormous bucket, taking down book after book , glancing at them, restoring them to the shelves or dumping them into the bucket. At last he came to three large volumes which Russell could recognize as the last surviving copy of Principia Mathematica. He took down one of the volumes, turned over a few pages , seemed puzzled for a moment by the curious symbolism , closed the volume , balanced it in his hand and hesitated …