Some egotism of this sort is inevitable, and I do not feel that it really needs justification. Good work is not done by ‘humble’ men. It is one of the first duties of a professor, for example, in any subject, to exaggerate a little both the importance of his subject and his own importance in it.
Dedication
To
致
JOHN LOMAS
约翰·洛马斯
who asked me to write it
Foreword
Like Einstein, he dressed to please himself: though, unlike Einstein, he diversified his casual clothing by a taste for expensive silk shirts.
As I had plenty of opportunities to realize in the future, Hardy had no faith in intuitions or impressions, his own or anyone else’s. The only way to assess someone’s knowledge, in Hardy’s view, was to examine him. That went for mathematics, literature, philosophy, politics, anything you like. If the man had bluffed and then wilted under the questions, that was his lookout. First things came first, in that brilliant and concentrated mind.
‘I still say to myself when I am depressed and find myself forced to listen to pompous and tiresome people, “Well, I have done one thing you could never have done, and that is to have collaborated with Littlewood and Ramanujan on something like equal terms.”’
Some of this self-consciousness wore off. He became competitive. As he says in the Apology: ‘I do not remember having felt, as a boy, any passion for mathematics, and such notions as I may have had of the career of a mathematician were far from noble. I thought of mathematics in terms of examinations and scholarships: I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively.’
The other unusual feature of his childhood was more mundane: but it meant the removal of all practical obstacles throughout his entire career. Hardy, with his limpid honesty, would have been the last man to be finicky on this matter. He knew what privilege meant, and he knew that he had possessed it. His family had no money, only a schoolmaster’s income, but they were in touch with the best educational advice of late nineteenth-century England. That particular kind of information has always been more significant in this country than any amount of wealth. The scholarships have been there all right, if one knew how to win them. There was never the slightest chance of the young Hardy being lost—as there was of the young Wells or the young Einstein. From the age of twelve he had only to survive, and his talents would be looked after.
At this point the young Einstein would have rebelled: he would either have left Cambridge or done no formal work for the next three years. But Hardy was born into the more intensely professional English climate (which has its merits as well as its demerits). After considering changing his subject to history, he had the sense to find a real mathematician to teach him.
Much of his life, though, he was happier than most of us. He had a great many friends, of surprisingly different kinds. These friends had to pass some of his private tests: they needed to possess a quality which he called ‘spin’ (this is a cricket term, and untranslatable: it implies a certain obliquity or irony of approach: of recent public figures, Macmillan and Kennedy would get high marks for spin, Churchill and Eisenhower not). But he was tolerant, loyal, extremely high-spirited, and in an undemonstrative way fond of his friends. I once was compelled to go and see him in the morning, which was always his set time for mathematical work. He was sitting at his desk, writing in his beautiful calligraphy. I murmured some commonplace politeness about hoping that I wasn’t disturbing him. He suddenly dissolved into his mischievous grin. ‘As you ought to be able to notice, the answer to that is that you are. Still, I’m usually glad to see you.’ In the sixteen years we knew each other, he didn’t say anything more demonstrative than that: except on his deathbed, when he told me that he looked forward to my visits.
With sardonic stoicism he says in the Apology— which for all its high spirits is a book of desperate sadness—that when a creative man has lost the power or desire to create—‘It is a pity but in that case he does not matter a great deal anyway, and it would be silly to bother about him.’ That is how he treated his personal life outside mathematics. Mathematics was his justification. It was easy to forget this, in the sparkle of his company: just as it was easy in the presence of Einstein’s moral passion to forget that to himself his justification was his search for the physical laws. Neither of those two ever forgot it. This was the core of their lives, from young manhood to death.
Again, this wasn’t false modesty: it was the judgment of a master who knew to an inch which of his work had value and which hadn’t.
At first glance, neither of them would have seemed the easiest of partners. It is hard to imagine either of them suggesting the collaboration in the first place. Yet one of them must have done. No one has any evidence how they set about it. Through their most productive period, they were not at the same university. Harald Bohr (brother of Niels Bohr, and himself a fine mathematician) is reported as saying that one of their principles was this: if one wrote a letter to the other, the recipient was under no obligation to reply to it, or even to read it.
Then, from about nine to one, unless he was giving a lecture, he worked at his own mathematics. Four hours creative work a day is about the limit for a mathematician, he used to say.
Hardy used to visit him, as he lay dying in hospital at Putney. It was on one of those visits that there happened the incident of the taxi-cab number. Hardy had gone out to Putney by taxi, as usual his chosen method of conveyance. He went into the room where Ramanujan was lying. Hardy, always inept about introducing a conversation, said, probably without a greeting, and certainly as his first remark: ‘I thought the number of my taxi-cab was 1729. It seemed to me rather a dull number.’ To which Ramanujan replied: ‘No, Hardy! No, Hardy! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.’
No one seemed to care—it was a gossipy college joke—that he had a large photograph of Lenin in his rooms. Hardy’s radicalism was somewhat unorganized, but it was real.
He would explain to an acquaintance that God, believing that Hardy expected the weather to change and give him a chance to work, counter-suggestibly arranged that the sky should remain cloudless.
‘It is never worth a first class man’s time to express a majority opinion. By definition, there are plenty of others to do that.’
That is why A Mathematician’s Apology is, if read with the textual attention it deserves, a book of haunting sadness. Yes, it is witty and sharp with intellectual high spirits: yes, the crystalline clarity and candour are still there: yes, it is the testament of a creative artist. But it is also, in an understated stoical fashion, a passionate lament for creative powers that used to be and that will never come again. I know nothing like it in the language: partly because most people with the literary gift to express such a lament don’t come to feel it: it is very rare for a writer to realize, with the finality of truth, that he is absolutely finished.
It was for that reason that some of his friends, including me, encouraged him to write the story of Bertrand Russell and Trinity in the 1914-18 war. People who didn’t know how depressed Hardy was thought the whole episode was now long over and ought not to be resurrected.
I used such persuasion as I had to get him to write another book, which he had in happier days promised me to do.
Chapter 1
IT is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings; there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.
I write about mathematics because, like any other mathematician who has passed sixty, I have no longer the freshness of mind, the energy, or the patience to carry on effectively with my proper job.
Chapter 2
Some egotism of this sort is inevitable, and I do not feel that it really needs justification. Good work is not done by ‘humble’ men. It is one of the first duties of a professor, for example, in any subject, to exaggerate a little both the importance of his subject and his own importance in it. A man who is always asking ‘Is what I do worth while?’ and ‘Am I the right person to do it?’ will always be ineffective himself and a discouragement to others. He must shut his eyes a little and think a little more of his subject and himself than they deserve. This is not too difficult: it is harder not to make his subject and himself ridiculous by shutting his eyes too tightly.
Chapter 3
‘I do what I do because it is the one and only thing that I can do at all well.
I am not suggesting that this is a defence which can be made by most people, since most people can do nothing at all well.
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.
Such a sacrifice could be justified only by economic necessity or age.
Chapter 4
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.
Chapter 5
‘There is nothing that I can do particularly well. I do what I do because it came my way. I really never had a chance of doing anything else.’ And this apology too I accept as conclusive. It is quite true that most people can do nothing well. If so, it matters very little what career they choose, and there is really nothing more to say about it. It is a conclusive reply, but hardly one likely to be made by a man with any pride; and I may assume that none of us would be content with it.
Chapter 6
‘the study of mathematics is, if an unprofitable, a perfectly harmless and innocent occupation’.
Chapter 7
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—
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.
Chapter 10
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.
Chapter 11
The best mathematics is serious as well as beautiful—‘important’ if you like, but the word is very ambiguous, and ‘serious’ expresses what I mean much better.
We may say, roughly, that a mathematical idea is ‘significant’ if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas. Thus a serious mathematical theorem, a theorem which connects significant ideas, is likely to lead to important advances in mathematics itself and even in other sciences.
Chapter 12
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.
Chapter 14
There is no doubt at all, then, of the ‘seriousness’ of either theorem. It is therefore the better worth remarking that neither theorem has the slightest ‘practical’ importance. In practical applications we are concerned only with comparatively small numbers; only stellar astronomy and atomic physics deal with ‘large’ numbers, and they have very little more practical importance, as yet, than the most abstract pure mathematics. I do not know what is the highest degree of accuracy which is ever useful to an engineer—we shall be very generous if we say ten significant figures.
So much for Euclid’s theorem; and, as regards Pythagoras’s, it is obvious that irrationals are uninteresting to an engineer, since he is concerned only with approximations, and all approximations are rational.
Chapter 16
‘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.’
Chapter 17
I could multiply examples, but this notion of ‘depth’ is an elusive one even for a mathematician who can recognize it, and I can hardly suppose that I could say anything more about it here which would be of much help to other readers.
Chapter 18
A chess problem also has unexpectedness, and a certain economy; it is essential that the moves should be surprising, and that every piece on the board should play its part. But the aesthetic effect is cumulative. It is essential also (unless the problem is too simple to be really amusing) that the key-move should be followed by a good many variations, each requiring its own individual answer.
Chapter 20
Of course we do not mean, when we say that physiology is useful, that most people ought to study physiology, but that the development of physiology by a handful of experts will increase the comfort of the majority.
Chapter 21
If the theory of numbers could be employed for any practical and obviously honourable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering, as physiology and even chemistry can, then surely neither Gauss nor any other mathematician would have been so foolish as to decry or regret such applications. But science works for evil as well as for good (and particularly, of course, in time of war); and both Gauss and lesser mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean.
I have been accused of taking this view myself. I once said that ‘a science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life’, and this sentence, written in 1915, has been quoted (for or against me) several times. It was of course a conscious rhetorical flourish, though one perhaps excusable at the time when it was written.
Chapter 22
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 our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it. A reader who does not like the philosophy can alter the language: it will make very little difference to my conclusions.
Chapter 26
The general conclusion, surely, stands out plainly enough. If useful knowledge is, as we agreed provisionally to say, knowledge which is likely, now or in the comparatively near future, to contribute to the material comfort of mankind, so that mere intellectual satisfaction is irrelevant, then the great bulk of higher mathematics is useless. Modern geometry and algebra, the theory of numbers, the theory of aggregates and functions, relativity, quantum mechanics—no one of them stands the test much better than another, and there is no real mathematician whose life can be justified on this ground. If this be the test, then Abel, Riemann, and Poincaré wasted their lives; their contribution to human comfort was negligible, and the world would have been as happy a place without them.
Chapter 27
and Hogben (who is as unsympathetic to what I and other mathematicians call mathematics as Whitehead is sympathetic) says that ‘without a knowledge of mathematics, the grammar of size and order, we cannot plan the rational society in which there will be leisure for all and poverty for none’ (and much more to the same effect).
Chapter 29
It is plain now that my life, for what it is worth, is finished, and that nothing I can do can perceptibly increase or diminish its value. It is very difficult to be dispassionate, but I count it a ‘success’; I have had more reward and not less than was due to a man of my particular grade of ability. I have held a series of comfortable and ‘dignified’ positions. I have had very little trouble with the duller routine of universities. I hate ‘teaching ‘, and have had to do very little, such teaching as I have done having been almost entirely supervision of research; I love lecturing, and have lectured a great deal to extremely able classes; and I have always had plenty of leisure for the researches which have been the one great permanent happiness of my life.
My choice was right, then, if what I wanted was a reasonably comfortable and happy life. But solicitors and stockbrokers and bookmakers often lead comfortable and happy lives, and it is very difficult to see how the world is the richer for their existence. Is there any sense in which I can claim that my life has been less futile than theirs? It seems to me again that there is only one possible answer: yes, perhaps, but, if so, for one reason only.
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 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.
Note
Thus (to take the latter point first), I must remember (a) that the organization of an entire population for war is only possible through scientific methods; (b) that science has greatly increased the power of propaganda, which is used almost exclusively for evil; and (c) that it has made ‘neutrality’ almost impossible or unmeaning, so that there are no longer ‘islands of peace’ from which sanity and restoration might spread out gradually after war.