Godfrey Harold Hardy

A Mathematician'sApology

BAALTIS PUBLISHING

Godfrey Harold Hardy

A Mathematician'sApology

foreword

It was a perfectly ordinary night at Christ’s high table,except that Hardy was dining as a guest. He had justreturned to Cambridge as Sadleirian professor, and Ihad heard something of him from young Cambridgemathematicians. They were delighted to have him back:he was a real mathematician, they said, not like thoseDiracs and Bohrs the physicists were always talkingabout: he was the purest of the pure. He was alsounorthodox, eccentric, radical, ready to talk about any-thing. This was 1931, and the phrase was not yet inEnglish use, but in later days they would have said thatin some inde�nable way he had star quality.

So, from lower down the table, I kept studying him.He was then in his early �fties: his hair was alreadygray, above skin so deeply sunburnt that it stayed akind of Red Indian bronze. His face was beautiful—highcheek bones, thin nose, spiritual and austere but capa-ble of dissolving into convulsions of internal gamin-likeamusement. He had opaque brown eyes, bright as abird’s—a kind of eye not uncommon among those witha gift for conceptual thought. Cambridge at that time

was full of unusual and distinguished faces—but eventhen, I thought that night, Hardy’s stood out.

I do not remember what he was wearing. It mayeasily have been a sports coat and grey �annels underhis gown. Like Einstein, he dressed to please himself:though, unlike Einstein, he diversi�ed his casual cloth-ing by a taste for expensive silk shirts.

As we sat round the combination-room table, drink-ing wine after dinner, someone said that Hardy wantedto talk to me about cricket. I had been elected only ayear before, but Christ’s was then a small college, andthe pastimes of even the junior fellows were soon iden-ti�ed. I was taken to sit by him. I was not introduced.He was, as I later discovered, shy and self-consciousin all formal actions, and had a dread of introductions.He just put his head down as it were in a butt of ac-knowledgment, and without any preamble what-everbegan:

‘You’re supposed to know something about cricket,aren’t you?’ Yes, I said, I knew a bit.

Immediately he began to put me through a moder-ately sti� viva. Did I play? What sort of performer wasI? I half-guessed that he had a horror of persons, thenprevalent in academic society, who devotedly studiedthe literature but had never played the game. I trottedout my credentials, such as they were. He appearedto �nd the reply partially reassuring, and went on tomore tactical questions. Whom should I have chosenas captain for the last test match a year before (in old-stylenums1930)? If the selectors had decided that Snowwas the man to save England, what would have been

2

my strategy and tactics? (‘You are allowed to act, if youare su�ciently modest, as non-playing captain.’) Andso on, oblivious to the rest of the table. He was quiteabsorbed.

As I had plenty of opportunities to realize in thefuture, Hardy had no faith in intuitions or impressions,his own or anyone else’s. The only way to assess some-one’s knowledge, in Hardy’s view, was to examine him.That went for mathematics, literature, philosophy, poli-tics, anything you like. If the man had blu�ed and thenwilted under the questions, that was his lookout. Firstthings came �rst, in that brilliant and concentratedmind.

That night in the combination-room, it was nec-essary to discover whether I should be tolerable asa cricket companion. Nothing else mattered. In theend he smiled with immense charm, with child-likeopenness, and said that Fenner’s (the university cricketground) next season might be bearable after all, withthe prospect of some reasonable conversation.

Thus, just as I owed my acquaintanceship withLloyd George to his passion for phrenology, I owed myfriendship with Hardy to having wasted a dispropor-tionate amount of my youth on cricket. I don’t knowwhat the moral is. But it was a major piece of luck forme. This was intellectually the most valuable friend-ship of my life. His mind, as I have just mentioned, wasbrilliant and concentrated: so much so that by his sideanyone else’s seemed a little muddy, a little pedestrianand confused. He wasn’t a great genius, as Einsteinand Rutherford were. He said, with his usual clarity,

3

that if the word meant anything he was not a geniusat all. At his best, he said, he was for a short time the�fth best pure mathematician in the world. Since hischaracter was as beautiful and candid as his mind, healways made the point that his friend and collaboratorLittlewood was an appreciably more powerful mathe-matician than he was, and that his protégé Ramanujanreally had natural genius in the sense (though not to theextent, and nothing like so e�ectively) that the greatestmathematicians had it.

People sometimes thought he was under-ratinghimself, when he spoke of these friends. It is true thathe was magnanimous, as far from envy as a man canbe: but I think one mistakes his quality if one doesn’taccept his judgment. I prefer to believe in his ownstatement in A Mathematician’s Apology, at the sametime so proud and so humble: ‘I still say to myselfwhen I am depressed and �nd myself forced to listento pompous and tiresome people, “Well, I have doneone thing you could never have done, and that is tohave collaborated with Littlewood and Ramanujan onsomething like equal terms.”’

In any case, his precise ranking must be left to thehistorians of mathematics (though it will be an almostimpossible job, since so much of his best work was donein collaboration). There is something else, though, atwhich he was clearly superior to Einstein or Rutherfordor any other great genius: and that is at turning anywork of the intellect, major or minor or sheer play, intoa work of art. It was that gift above all, I think, whichmade him, almost without realizing it, purvey such

4

intellectual delight. When A Mathematician’s Apologywas �rst published, Graham Greene in a review wrotethat along with Henry James’s notebooks, this was thebest account of what it was like to be a creative artist.Thinking about the e�ect Hardy had on all those roundhim, I believe that is the clue.

He was born, in 1877, into a modest professionalfamily. His father was Bursar and Art Master at Cran-leigh, then a minor public (English for private) school.His mother had been senior mistress at the LincolnTraining College for teachers. Both were gifted andmathematically inclined. In his case, as in that of mostmathematicians, the gene pool doesn’t need searchingfor. Much of his childhood, unlike Einstein’s, was typi-cal of a future mathematician’s. He was demonstratinga formidably high i.q. as soon as, or before, he learnedto talk. At the age of two he was writing down numbersup to millions (a common sign of mathematical ability).When he was taken to church he amused himself byfactorizing the numbers of the hymns: he played withnumbers from that time on, a habit which led to thetouching scene at Ramanujan’s sick-bed: the scene iswell known, but later on I shall not be able to resistrepeating it.

It was an enlightened, cultivated, highly literateVictorian childhood. His parents were probably a littleobsessive, but also very kind. Childhood in such a Vic-torian family was as gentle a time as anything we couldprovide, though probably intellectually somewhat moreexacting. His was unusual in just two respects. In the�rst place, he su�ered from an acute self-consciousness

5

at an unusually early age, long before he was twelve.His parents knew he was prodigiously clever, and sodid he. He came top of his class in all subjects. But, asthe result of coming top of his class, he had to go infront of the school to receive prizes: and that he couldnot bear. Dining with me one night, he said that hedeliberately used to try to get his answers wrong so asto be spared this intolerable ordeal. His capacity fordissimulation, though, was always minimal: he got theprizes all the same.

Some of this self-consciousness wore o�. He be-came competitive. As he says in the Apology: ‘I do notremember having felt, as a boy, any passion for mathe-matics, and such notions as I may have had of the careerof a mathematician were far from noble. I thought ofmathematics in terms of examinations and scholarships:I wanted to beat other boys, and this seemed to be theway in which I could do so most decisively.’ Neverthe-less, he had to live with an over-delicate nature. Heseems to have been born with three skins too few. Un-like Einstein, who had to subjugate his powerful egoin the study of the external world before he could at-tain his moral stature, Hardy had to strengthen an egowhich wasn’t much protected. This at times in later lifemade him self-assertive (as Einstein never was) whenhe had to take a moral stand. On the other hand, it gavehim his introspective insight and beautiful candor, sothat he could speak of himself with absolute simplicity(as Einstein never could).

I believe this contradiction, or tension, in his tem-perament was linked with a curious tic in his behavior.

6

He was the classical anti-narcissist. He could not en-dure having his photograph taken: so far as I know,there are only �ve snapshots in existence. He wouldnot have any looking glass in his rooms, not even ashaving mirror. When he went to a hotel, his �rst ac-tion was to cover all the looking-glasses with towels.This would have been odd enough, if his face had beenlike a gargoyle: super�cially it might seem odder, sinceall his life he was good-looking quite out of the ordi-nary. But, of course, narcissism and anti-narcissismhave nothing to do with looks as outside observers seethem.

This behavior seems eccentric, and indeed it was.Between him and Einstein, though, there was a dif-ference in kind. Those who spent much time withEinstein—such as Infeld—found him grow stranger, lesslike themselves, the longer they knew him. I am cer-tain that I should have felt the same. With Hardy theopposite was true. His behavior was often di�erent,bizarrely so, from ours: but it came to seem a kindof superstructure set upon a nature which wasn’t allthat di�erent from our own, except that it was moredelicate, less padded, �ner-nerved.

The other unusual feature of his childhood wasmore mundane: but it meant the removal of all practi-cal obstacles throughout his entire career. Hardy, withhis limpid honesty, would have been the last man tobe �nicky on this matter. He knew what privilegemeant, and he knew that he had possessed it. His fam-ily had no money, only a schoolmaster’s income, butthey were in touch with the best educational advice of

7

late nineteenth-century England. That particular kindof information has always been more signi�cant inthis country than any amount of wealth. The scholar-ships have been there all right, if one knew how towin them. There was never the slightest chance of theyoung Hardy being lost—as there was of the youngWells or the young Einstein. From the age of twelve hehad only to survive, and his talents would be lookedafter.

At twelve, in fact, he was given a scholarship atWinchester, then and for long afterwards the best math-ematical school in England, simply on the strength ofsome mathematical work he had done at Cranleigh. (In-cidentally, one wonders if any great school could be soelastic nowadays?) There he was taught mathematicsin a class of one: in classics he was as good as the othertop collegers. Later, he admitted that he had been well-educated, but he admitted it reluctantly. He dislikedthe school, except for its classes. Like all Victorian pub-lic schools, Winchester was a pretty rough place. Henearly died one winter. He envied Littlewood in hiscared-for home as a day boy at St Paul’s or other friendsat our free-and-easy grammar schools. He never wentnear Winchester after he had left it: but he left it, withthe inevitability of one who had got on to the righttramlines, with an open scholarship to Trinity.

He had one curious grievance against Winchester.He was a natural ball-games player with a splendideye. In his �fties he could usually beat the universitysecond string at real tennis, and in his sixties I saw himbring o� startling shots in the cricket nets. Yet he had

8

not had an hour’s coaching at Winchester: his methodwas defective: if he had been coached, he thought, hewould have been a really good batsman, not quite �rst-class, but not too far away. Like all his judgments onhimself, I believe that one is quite true. It is strangethat, at the zenith of Victorian games-worship, such atalent was utterly missed. I suppose no one thought itworth looking for in the school’s top scholar, so frailand sickly, so defensively shy.

It would have been natural for a Wykehamist of hisperiod to go to New College. That wouldn’t have mademuch di�erence to his professional career (though,since he always liked Oxford better than Cambridge,he might have stayed there all his life, and some of uswould have missed a treat). He decided to go to Trinityinstead, for a reason that he describes, humorously butwith his usual undecorated truth, in the Apology. “Iwas about �fteen when (in a rather odd way) my ambi-tions took a sharper turn. There is a book by ‘Alan StAubyn’ (actually Mrs Frances Marshall) called A Fellowof Trinity, one of a series dealing with what is supposedto be Cambridge college life. . . There are two heroes,a primary hero called Flowers, who is almost whollygood, and a secondary hero, a much weaker vessel,called Brown. Flowers and Brown �nd many dangersin university life. . . Flowers survives all these troubles,is Second Wrangler and succeeds automatically to a Fel-lowship (as I suppose he would have done then). Brownsuccumbs, ruins his parents, takes to drink, is savedfrom delirium tremens during a thunderstorm only bythe prayers of the Junior Dean, has much di�culty in

9

obtaining even an Ordinary Degree, and ultimately be-comes a missionary. The friendship is not shatteredby these unhappy events, and Flowers’s thoughts strayto Brown, with a�ectionate pity, as he drinks port andeats walnuts for the �rst time in Senior CombinationRoom.

‘Now Flowers was a decent enough fellow (so faras “Alan St Aubyn” could draw one), but even my unso-phisticated mind refused to accept him as clever. If hecould do these things, why not I? In particular, the �nalscene in Combination Room fascinated me completely,and from that time, until I obtained one, mathematicsmeant to me primarily a Fellowship of Trinity.’

Which he duly obtained, after getting the highestplace in the Mathematical Tripos Part II, at the ageof 22. On the way, there were two minor vicissitudes.The �rst was theological, in the high Victorian manner.Hardy had decided—I think before he left Winchester—that he did not believe in God. With him, this wasa black-and-white decision, as sharp and clear as allother concepts in his mind. Chapel at Trinity was com-pulsory. Hardy told the Dean, no doubt with his ownkind of shy certainty, that he could not conscientiouslyattend. The Dean, who must have been a jack-in-o�ce,insisted that Hardy should write to his parents and tellthem so. They were orthodox religious people, and theDean knew, and Hardy knew much more, that the newswould give them pain—pain such as we, seventy yearslater, cannot easily imagine.

Hardy struggled with his conscience. He wasn’tworldly enough to slip the issue. He wasn’t even worldly

10

enough—he told me one afternoon at Fenner’s, for thewound still rankled—to take the advice of more sophis-ticated friends, such as George Trevelyan and DesmondMacCarthy, who would have known how to handle thematter. In the end he wrote the letter. Partly becauseof that incident, his religious disbelief remained openand active ever after. He refused to go into any collegechapel even for formal business, like electing a master.He had clerical friends, but God was his personal enemy.In all this there was an echo of the nineteenth century:but one would be wrong, as always with Hardy, not totake him at his word.

Still, he turned it into high jinks. I remember, oneday in the thirties, seeing him enjoy a minor triumph.It happened in a Gentlemen v. Players match at Lord’s.It was early in the morning’s play, and the sun wasshining over the pavilion. One of the batsmen, facingthe Nursery end, complained that he was unsighted bya re�ection from somewhere unknown. The umpires,puzzled, padded round by the sight-screen. Motor-cars?No. Windows? None on that side of the ground. At last,with justi�able triumph, an umpire traced the re�ec-tion down—it came from a large pectoral cross reposingon the middle of an enormous clergyman. Politely theumpire asked him to take it o�. Close by, Hardy wasdoubled up in Mephistophelian delight. That lunchtime, he had no leisure for eating: he was writing post-cards (postcards and telegrams were his favorite meansof communication) to each of his clerical friends.

But in his war against God and God’s surrogates,victory was not all on one side. On a quiet and lovely

11

May evening at Fenner’s, round about the same period,the chimes of six o’clock fell across the ground. ‘It’srather unfortunate,’ said Hardy simply, ‘that some ofthe happiest hours of my life should have been spentwithin sound of a Roman Catholic church.’

The second minor upset of his undergraduate yearswas professional. Almost since the time of Newton,and all through the nineteenth century, Cambridge hadbeen dominated by the examination for the old Math-ematical Tripos. The English have always had morefaith in competitive examinations than any other peo-ple (except perhaps the Imperial Chinese): they haveconducted these examinations with traditional justice:but they have often shown remarkable woodenness indeciding what the examinations should be like. That is,incidentally, true to this day. It was certainly true of theMathematical Tripos in its glory. It was an examinationin which the questions were usually of considerablemechanical di�culty—but unfortunately did not giveany opportunity for the candidate to show mathemati-cal imagination or insight or any quality that a creativemathematician needs. The top candidates (the Wran-glers —a term which still survives, meaning a FirstClass) were arranged, on the basis of marks, in strictnumerical order. Colleges had celebrations when oneof their number became Senior Wrangler: the �rst twoor three Wranglers were immediately elected Fellows.

It was all very English. It had only one disadvan-tage, as Hardy pointed out with his polemic clarity, assoon as he had become an eminent mathematician andwas engaged, together with his tough ally Littlewood,

12

in getting the system abolished: it had e�ectively ru-ined serious mathematics in England for a hundredyears.

In his �rst term at Trinity, Hardy found himselfcaught in this system. He was to be trained as a race-horse, over a course of mathematical exercises whichat nineteen he knew to be meaningless. He was sent toa famous coach, to whom most potential Senior Wran-glers went. This coach knew all the obstacles, all thetricks of the examiners, and was sublimely uninter-ested in the subject itself. At this point the youngEinstein would have rebelled: he would either have leftCambridge or done no formal work for the next threeyears. But Hardy was born into the more intenselyprofessional English climate (which has its merits aswell as its demerits). After considering changing hissubject to history, he had the sense to �nd a real math-ematician to teach him. Hardy paid him a tribute inthe Apology: ‘My eyes were �rst opened by ProfessorLove, who taught me for a few terms and gave memy �rst serious conception of analysis. But the greatdebt which I owe to him—he was, after all, primarilyan applied mathematician—was his advice to read Jor-dan’s famous Cours d’Analyse: and I shall never forgetthe astonishment with which I read that remarkablework, the �rst inspiration for so many mathematiciansof my generation, and learned for the �rst time as Iread it what mathematics really meant. From that timeonwards I was in my way a real mathematician, withsound mathematical ambitions and a genuine passionfor mathematics.’

13

He was Fourth Wrangler in 1898. This faintly ir-ritated him, he used to confess. He was enough of anatural competitor to feel that, though the race wasridiculous, he ought to have won it. In 1900 he took Partii of the Tripos, a more respect-worthy examination,and got his right place and his Fellowship.

From that time on, his life was in essence settled. Heknew his purpose, which was to bring rigor into Englishmathematical analysis. He did not deviate from theresearches which he called ‘the one great permanenthappiness of my life. There were no anxieties aboutwhat he should do. Neither he nor anyone else doubtedhis great talent. He was elected to the Royal Society atthirty-three.

In many senses, then, he was unusually lucky. Hedid not have to think about his career. From the timehe was twenty-three he had all the leisure that a mancould want, and as much money as he needed. A bach-elor don in Trinity in the 1900’s was comfortably o�.Hardy was sensible about money, spent it when hefelt impelled (sometimes for singular purposes, such as�fty-mile taxi-rides), and otherwise was not at all un-worldly about in-vestments. He played his games andindulged his eccentricities. He was living in some ofthe best intellectual company in the world—G.E. Moore,Whitehead, Bertrand Russell, Trevelyan, the high Trin-ity society which was shortly to �nd its artistic com-plement in Bloomsbury. (Hardy himself had links withBloomsbury, both of personal friendship and of sym-pathy.) In a brilliant circle, he was one of the most

14

brilliant young men—and, in a quiet way, one of themost irrepressible.

I will anticipate now what I shall say later. His liferemained the life of a brilliant young man until he wasold: so did his spirit: his games, his interests, kept thelightness of a young don’s. And, like many men whokeep a young man’s interests into their sixties, his lastyears were the darker for it.

Much of his life, though, he was happier than mostof us. He had a great many friends, of surprisinglydi�erent kinds. These friends had to pass some of hisprivate tests: they needed to possess a quality which hecalled ‘spin’ (this is a cricket term, and untranslatable:it implies a certain obliquity or irony of approach: ofrecent public �gures, Macmillan and Kennedy wouldget high marks for spin, Churchill and Eisenhower not).But he was tolerant, loyal, extremely high-spirited, andin an undemonstrative way fond of his friends. I oncewas compelled to go and see him in the morning, whichwas always his set time for mathematical work. He wassitting at his desk, writing in his beautiful calligraphy. Imurmured some commonplace politeness about hopingthat I wasn’t disturbing him. He suddenly dissolvedinto his mischievous grin. ‘As you ought to be ableto notice, the answer to that is that you are. Still, I’musually glad to see you.’ In the sixteen years we kneweach other, he didn’t say anything more demonstrativethan that: except on his deathbed, when he told methat he looked forward to my visits.

I think my experience was shared by most of hisclose friends. But he had, scattered through his life, two

15

or three other relationships, di�erent in kind. Thesewere intense a�ections, absorbing, non-physical butexalted. The one I knew about was for a young manwhose nature was as spiritually delicate as his own. I be-lieve, though I only picked this up from chance remarks,that the same was true of the others. To many peopleof my generation, such relationships would seem ei-ther unsatisfactory or impossible. They were neitherthe one nor the other; and, unless one takes them forgranted, one doesn’t begin to understand the tempera-ment of men like Hardy (they are rare, but not as rareas white rhinoceroses), nor the Cambridge society ofhis time. He didn’t get the satisfactions that most of uscan’t help �nding: but he knew himself unusually well,and that didn’t make him unhappy. His inner life washis own, and very rich. The sadness came at the end.Apart from his devoted sister, he was left with no oneclose to him.

With sardonic stoicism he says in the Apology—which for all its high spirits is a book of desperatesadness—that when a creative man has lost the poweror desire to create—’It is a pity but in that case he doesnot matter a great deal anyway, and it would be sillyto bother about him.’ That is how he treated his per-sonal life outside mathematics. Mathematics was hisjusti�cation. It was easy to forget this, in the sparkleof his company: just as it was easy in the presenceof Einstein’s moral passion to forget that to himselfhis justi�cation was his search for the physical laws.Neither of those two ever forgot it. This was the coreof their lives, from young manhood to death.

16

Hardy, unlike Einstein, did not make a quick start.His early papers, between 1900 and 1911, were goodenough to get him into the Royal Society and win himan international reputation: but he did not regard themas important. Again, this wasn’t false modesty: it wasthe judgment of a master who knew to an inch whichof his work had value and which hadn’t.

In 1911 he began a collaboration with Littlewoodwhich lasted thirty-�ve years. In 1913 he discoveredRamanujan and began another collaboration. All hismajor work was done in these two partnerships, mostof it in the one with Littlewood, the most famous collab-oration in the history of mathematics. There has beennothing like it in any science, or, so far as I know, inany other �eld of creative activity. Together they pro-duced nearly a hundred papers, a good many of them‘in the Bradman class’. Mathematicians not intimatewith Hardy in his later years, nor with cricket, keeprepeating that his highest term of praise was ‘in theHobbs class’. It wasn’t: very reluctantly, because Hobbswas one of his pets, he had to alter the order of merit. Ionce had a postcard from him, probably in 1938, saying‘Bradman is a whole class above any batsman who hasever lived: if Archimedes, Newton and Gauss remain inthe Hobbs class, I have to admit the possibility of a classabove them, which I �nd di�cult to imagine. They hadbetter be moved from now on into the Bradman class.’

The Hardy-Littlewood researches dominated En-glish pure mathematics, and much of world pure math-ematics, for a generation. It is too early to say, somathematicians tell me, to what extent they altered

17

the course of mathematical analysis: nor how in�uen-tial their work will appear in a hundred years. Of itsenduring value there is no question.

Theirs was, as I have said, the greatest of all col-laborations. But no one knows how they did it: unlessLittlewood tells us, no one will ever know. I have al-ready given Hardy’s judgment that Littlewood was themore powerful mathematician of the two: Hardy oncewrote that he knew of ‘no one else who could com-mand such a combination of insight, technique andpower.’ Littlewood was and is a more normal man thanHardy, just as interesting and probably more complex.He never had Hardy’s taste for a kind of re�ned intel-lectual �amboyance, and so was less in the center ofthe academic scene. This led to jokes from Europeanmathematicians, such as that Hardy had invented himso as to take the blame in case there turned out any-thing wrong with one of their theorems. In fact, he is aman of at least as obstinate an individuality as Hardyhimself.

At �rst glance, neither of them would have seemedthe easiest of partners. It is hard to imagine either ofthem suggesting the collaboration in the �rst place. Yetone of them must have done. No one has any evidencehow they set about it. Through their most productiveperiod, they were not at the same university. HaraldBohr (brother of Niels Bohr, and himself a �ne mathe-matician) is reported as saying that one of their prin-ciples was this: if one wrote a letter to the other, therecipient was under no obligation to reply to it, or evento read it.

18

I can’t contribute anything. Hardy talked to me,over a period of many years, on almost every con-ceivable subject, except the collaboration. He said, ofcourse, that it had been the major fortune of his creativecareer: he spoke of Littlewood in the terms I have given:but he never gave a hint of their procedures. I didn’tknow enough mathematics to understand their papers,but I picked up some of their language. If he had let slipanything about their methods, I don’t think I shouldhave missed it. I am fairly certain that the secrecy—quite uncharacteristic of him in matters which to mostwould seem more intimate—was deliberate.

About his discovery of Ramanujan, he showed nosecrecy at all. It was, he wrote, the one romantic inci-dent in his life: anyway, it is an admirable story, andone which showers credit on nearly everyone (withtwo exceptions) in it. One morning early in 1913, hefound, among the letters on his breakfast table, a largeuntidy envelope decorated with Indian stamps. Whenhe opened it, he found sheets of paper by no meansfresh, on which, in a non-English holograph, were lineafter line of symbols. Hardy glanced at them withoutenthusiasm. He was by this time, at the age of thirty-six, a world famous mathematician: and world famousmathematicians, he had already discovered, are unusu-ally exposed to cranks. He was accustomed to receivingmanuscripts from strangers, proving the prophetic wis-dom of the Great Pyramid, the revelations of the Eldersof Zion, or the cryptograms that Bacon had inserted inthe plays of the so-called Shakespeare.

19

So Hardy felt, more than anything, bored. He glancedat the letter, written in halting English, signed by anunknown Indian, asking him to give an opinion of thesemathematical discoveries. The script appeared to con-sist of theorems, most of them wild or fantastic looking,one or two already well-known, laid out as though theywere original. There were no proofs of any kind. Hardywas not only bored, but irritated. It seemed like a cu-rious kind of fraud. He put the manuscript aside, andwent on with his day’s routine. Since that routine didnot vary throughout his life, it is possible to reconstructit. First he read The Times over his breakfast. This hap-pened in January, and if there were any Australiancricket scores, he would start with them, studied withclarity and intense attention.

Maynard Keynes, who began his career as a mathe-matician and who was a friend of Hardy’s, once scoldedhim: if he had read the stock exchange quotationshalf an hour each day with the same concentrationhe brought to the cricket scores, he could not havehelped becoming a rich man.

Then, from about nine to one, unless he was givinga lecture, he worked at his own mathematics. Fourhours creative work a day is about the limit for a math-ematician, he used to say. Lunch, a light meal, in hall.After lunch he loped o� for a game of real tennis inthe university court. (If it had been summer, he wouldhave walked down to Fenner’s to watch cricket.) Inthe late afternoon, a stroll back to his rooms. Thatparticular day, though, while the timetable wasn’t al-tered, internally things were not going according to

20

plan. At the back of his mind, getting in the way of hiscomplete pleasure in his game, the Indian manuscriptnagged away. Wild theorems. Theorems such as hehad never seen before, nor imagined. A fraud of ge-nius? A question was forming itself in his mind. Asit was Hardy’s mind, the question was forming itselfwith epigrammatic clarity: is a fraud of genius moreprobable than an unknown mathematician of genius?Clearly the answer was no. Back in his rooms in Trin-ity, he had another look at the script. He sent word toLittlewood (probably by messenger, certainly not bytelephone, for which, like all mechanical contrivancesincluding fountain pens, he had a deep distrust) thatthey must have a discussion after hall.

When the meal was over, there may have been aslight delay. Hardy liked a glass of wine, but, despitethe glorious vistas of ‘Alan St. Aubyn’ which had �redhis youthful imagination, he found he did not reallyenjoy lingering in the combination-room over port andwalnuts. Littlewood, a good deal more homme moyensensuel, did. So there may have been a delay. Anyway,by nine o’clock or so they were in one of Hardy’s rooms,with the manuscript stretched out in front of them.

That is an occasion at which one would have likedto be present. Hardy, with his combination of remorse-less clarity and intellectual panache (he was very En-glish, but in argument he showed the characteristicsthat Latin minds have often assumed to be their own):Littlewood, imaginative, powerful, humorous. Appar-ently it did not take them long. Before midnight theyknew, and knew for certain. The writer of these manuscripts

21

was a man of genius. That was as much as they couldjudge, that night. It was only later that Hardy decidedthat Ramanujan was, in terms of natural mathematicalgenius, in the class of Gauss and Euler: but that hecould not expect, because of the defects of his educa-tion, and because he had come on the scene too late inthe line of mathematical history, to make a contributionon the same scale.

It all sounds easy, the kind of judgment great math-ematicians should have been able to make. But I men-tioned that there were two persons who do not comeout of the story with credit. Out of chivalry Hardy con-cealed this in all that he said or wrote about Ramanujan.The two people concerned have now been dead, how-ever, for many years, and it is time to tell the truth. Itis simple. Hardy was not the �rst eminent mathemati-cian to be sent the Ramanujan manuscripts. There hadbeen two before him, both English, both of the high-est professional standard. They had each returned themanuscripts without comment. I don’t think historyrelates what they said, if anything, when Ramanujanbecame famous. Anyone who has been sent unsolicitedmaterial will have a sneaking sympathy with them.

Anyway, the following day Hardy went into action.Ramanujan must be brought to England, he decided.Money was not a major problem. Trinity has usuallybeen good at supporting unorthodox talent (the collegedid the same for Kapitsa a few years later). Once Hardywas determined, no human agency could have stoppedRamanujan, but they needed a certain amount of helpfrom a superhuman one.

22

Ramanujan turned out to be a poor clerk in Madras,living with his wife on twenty pounds a year. But hewas also a Brahmin, unusually strict about his religiousobservances, with a mother who was even stricter. Itseemed impossible that he could break the proscrip-tions and cross the water. Fortunately his mother hadthe highest respect for the goddess of Namakkal. Onemorning Ramanujan’s mother made a startling announce-ment. She had had a dream on the previous night, inwhich she saw her son seated in a big hall among agroup of Europeans, and the goddess of Namakkal hadcommanded her not to stand in the way of her sonful�lling his life’s purpose. This, say Ramanujan’s In-dian biographers, was a very agreeable surprise to allconcerned.

In 1914 Ramanujan arrived in England. So far asHardy could detect (though in this respect I should nottrust his insight far) Ramanujan, despite the di�cul-ties of breaking the caste proscriptions, did not believemuch in theological doctrine, except for a vague pan-theistic benevolence, any more than Hardy did himself.But he did certainly believe in ritual. When Trinityput him up in college—within four years he became aFellow— there was no ‘Alan St. Aubyn’ apolausticityfor him at all. Hardy used to �nd him ritually changedinto his pyjamas, cooking vegetables rather miserablyin a frying pan in his own room.

Their association was a strangely touching one.Hardy did not forget that he was in the presence ofgenius: but genius that was, even in mathematics, al-most untrained. Ramanujan had not been able to enter

23

Madras University because he could not matriculate inEnglish. According to Hardy’s report, he was alwaysamiable and good-natured, but no doubt he sometimesfound Hardy’s conversation outside mathematics morethan a little ba�ing. He seems to have listened with apatient smile on his good, friendly, homely face. Eveninside mathematics they had to come to terms withthe di�erence in their education. Ramanujan was self-taught: he knew nothing of the modern rigor: in asense he didn’t know what a proof was. In an unchar-acteristically sloppy moment, Hardy once wrote that ifhe had been better educated, he would have been lessRamanujan. Coming back to his ironic senses, Hardylater corrected himself and said that the statement wasnonsense. If Ramanujan had been better educated, hewould have been even more wonderful than he was.In fact, Hardy was obliged to teach him some formalmathematics as though Ramanujan had been a schol-arship candidate at Winchester. Hardy said that thiswas the most singular experience of his life: what didmodern mathematics look like to someone who hadthe deepest insight, but who had literally never heardof most of it?

Anyway, they produced together �ve papers ofthe highest class, in which Hardy showed supremeoriginality of his own (more is known of the details ofthis collaboration than of the Hardy-Littlewood one).Generosity and imagination were, for once, rewardedin full.

This is a story of human virtue. Once people hadstarted behaving well, they went on behaving better.

24

It is good to remember that England gave Ramanujansuch honors as were possible. The Royal Society electedhim a Fellow at the age of thirty (which, even for amathematician, is very young). Trinity also electedhim a Fellow in the same year. He was the �rst Indianto be given either of these distinctions. He was amiablygrateful. But he soon became ill. It was di�cult, inwar-time, to move him to a kinder climate.

Hardy used to visit him, as he lay dying in hospitalat Putney. It was on one of those visits that there hap-pened the incident of the taxi-cab number. Hardy hadgone out to Putney by taxi, as usual his chosen methodof conveyance. He went into the room where Ramanu-jan was lying. Hardy, always inept about introducinga conversation, said, probably without a greeting, andcertainly as his �rst remark: ‘I thought the number ofmy taxicab was 1729. It seemed to me rather a dullnumber.’ To which Ramanujan replied: ‘No, Hardy!No, Hardy! It is a very interesting number. It is thesmallest number expressible as the sum of two cubesin two di�erent ways.’

That is the exchange as Hardy recorded it. It mustbe substantially accurate. He was the most honest ofmen; and further, no one could possibly have inventedit.

Ramanujan died of tuberculosis, back in Madras,two years after the war. As Hardy wrote in the Apol-ogy, in his roll-call of mathematicians: ‘Galois died attwenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. . . I do not know an instance of

25

a major mathematical advance initiated by a man past�fty.’

If it had not been for the Ramanujan collaboration,the 1914-18war would have been darker for Hardy thanit was. But it was dark enough. It left a wound whichreopened in the second war. He was a man of radicalopinions all his life. His radicalism, though, was tingedwith the turn-of-the-century enlightenment. To peopleof my generation, it sometimes seemed to breathe alighter, more innocent air, than the one we knew.

Like many of his Edwardian intellectual friends, hehad a strong feeling for Germany. Germany had, afterall, been the great educating force of the nineteenthcentury. To Eastern Europe, to Russia, to the UnitedStates, it was the German universities which had taughtthe meaning of research. Hardy hadn’t much use forGerman philosophy or German literature: his tasteswere too classical for that. But in most respects theGerman culture, including its social welfare, appearedto him higher than his own.

Unlike Einstein, who had a much more ruggedsense of political existence, Hardy did not know muchof Wilhelmine Germany at �rst hand. And, though hewas the least vain of people, he would have been lessthan human if he had not enjoyed being more appre-ciated in Germany than in his own country. There isa pleasant anecdote, dating from this period, in whichHilbert, one of the greatest of German mathematicians,heard that Hardy lived in a not specially agreeable set ofrooms in Trinity (actually in Whewell’s Court). Hilbertpromptly wrote in measured terms to the Master, point-

26

ing out that Hardy was the best mathematician, notonly in Trinity but in England, and should thereforehave the best rooms.

So Hardy, like Russell and many of the high Cam-bridge intelligentsia, did not believe that the war shouldhave been fought. Further, with his ingrained distrustof English politicians, he thought the balance of wrongwas on the English side. He could not �nd a satisfac-tory basis for conscientious objection; his intellectualrigor was too strong for that. In fact, he volunteeredfor service under the Derby scheme, and was rejectedon medical grounds. But he felt increasingly isolatedin Trinity, much of which was vociferously bellicose.

Russell was dismissed from his lectureship, in cir-cumstances of overheated complexity (Hardy was towrite the only detailed account of the case a quarter ofa century later, in order to comfort himself in anotherwar). Hardy’s close friends were away at the war. Little-wood was doing ballistics as a Second Lieutenant in theRoyal Artillery. Owing to his cheerful indi�erence hehad the distinction of remaining a Second Lieutenantthrough the four years of war. Their collaboration wasinterfered with, though not entirely stopped. It was thework of Ramanujan which was Hardy’s solace duringthe bitter college quarrels.

I sometimes thought he was, for once, less than fairto his colleagues. Some were pretty crazed, as men arein war-time. But some were long-su�ering and tried tokeep social manners going. After all, it was a triumphof academic uprightness that they should have electedhis protege Ramanujan, at a time when Hardy was only

27

just on speaking terms with some of the electors, andnot at all with others.

Still, he was harshly unhappy. As soon as he could,he left Cambridge. He was o�ered a chair at Oxfordin 1919: and immediately walked into the happiesttime of his life. He had already done great work withRamanujan and Littlewood, but now the collaborationwith Littlewood rose to its full power. Hardy was, inNewton’s phrase, ‘in the prime of his age for invention,’and this came in his early forties, unusually late for amathematician.

Coming so late, this creative surge gave him thefeeling, more important to him than to most men, oftimeless youth. He was living the young man’s lifewhich was �rst nature to him. He played more realtennis, and got steadily better at it (real tennis wasan expensive game and took a largish slice out of aprofessorial income). He made a good many visits toAmerican universities, and loved the country. He wasone of the few Englishmen of his time who was fond,to an extent approximately equal, of the United Statesand the Soviet Union. He was certainly the only En-glishman of his or any other time who wrote a serioussuggestion to the Baseball Commissioners, proposing atechnical emendation to one of the rules. The twenties,for him and for most liberals of his generation, wasa false dawn. He thought the misery of the war wasswept away into the past.

He was at home in New College as he had neverbeen in Cambridge. The warm domestic conversationalOxford climate was good for him. It was there, in a

28

college at that time small and intimate, that he perfectedhis own style of conversation. There was companyeager to listen to him after hall. They could take hiseccentricities. He was not only a great and good man,they realized, but an entertaining one. If he wantedto play conversational games, or real (though bizarre)games on the cricket �eld, they were ready to act asfoils. In a casual and human fashion, they made a fussof him. He had been admired and esteemed before, butnot made a fuss of in that fashion.

No one seemed to care—it was a gossipy collegejoke—that he had a large photograph of Lenin in hisrooms. Hardy’s radicalism was somewhat unorganized,but it was real. He had been born, as I have explained,into a professional family: almost all his life was spentamong the haute bourgeoisie: but in fact he behavedmuch more like an aristocrat, or more exactly like oneof the romantic projections of an aristocrat. Some ofthis attitude, perhaps, he had picked up from his friendBertrand Russell. But most of it was innate. Underneathhis shyness, he just didn’t give a damn.

He got on easily, without any patronage, with thepoor, the unlucky and di�dent, those who were hand-icapped by race (it was a symbolic stroke of fate thathe discovered Ramanujan). He preferred them to thepeople whom he called the large bottomed: the descrip-tion was more psychological than anatomical, thoughthere was a famous nineteenth-century Trinity apho-rism by Adam Sedgwick: ‘No one ever made a successin this world without a large bottom.’ To Hardy thelarge bottomed were the con�dent, booming, imperial-

29

ist bourgeois English. The designation included mostbishops, headmasters, judges, and all politicians, withthe single exception of Lloyd George.

Just to show his allegiances, he accepted one publico�ce. For two years (1924-26) he was President of theAssociation of Scienti�c Workers. He said sarcasticallythat he was an odd choice, being ‘the most unpracti-cal member of the most unpractical profession in theworld.’ But in important things he was not so unprac-tical. He was deliberately standing up to be counted.When, much later, I came to work with Frank Cousins,it gave me a certain quiet pleasure to re�ect that I hadhad exactly two friends who had held o�ce in the TradeUnion movement, him and G.H. Hardy.

That late, not quite Indian, summer in Oxford inthe twenties was so happy that people wondered thathe ever returned to Cambridge. Which he did in 1931.I think there were two reasons. First and most decisive,he was a great professional. Cambridge was still thecenter of English mathematics, and the senior math-ematical chair there was the correct place for a pro-fessional. Secondly, and rather oddly, he was thinkingabout his old age. Oxford colleges, in many ways sohuman and warm, are ruthless with the old: if he re-mained at New College he would be turned out of hisrooms as soon as he retired, under the age limit, fromhis professorship. Whereas, if he returned to Trinity,he could stay in college until he died. That is in e�ectwhat he managed to do.

When he came back to Cambridge—which was thetime that I began to know him—he was in the afterglow

30

of his great period. He was still happy. He was stillcreative, not so much as in the twenties, but enough tomake him feel that the power was still there. He wasas spirited as he had been at New College. So we hadthe luck to see him very nearly at his best.

In the winters, after we had become friendly, wegave each other dinner in our respective colleges once afortnight. When summer came, it was taken for grantedthat we should meet at the cricket ground. Except onspecial occasions he still did mathematics in the morn-ing, and did not arrive at Fenner’s until after lunch. Heused to walk round the cinderpath with a long, lop-ing, dumping-footed stride (he was a slight spare man,physically active even in his late �fties, still playing realtennis) head down, hair, tie, sweaters, papers all �ow-ing , a �gure that caught everyone’s eyes. ‘There goesa Greek poet, I’ll be bound,’ once said some cheerfulfarmer as Hardy passed the score-board. He made forhis favorite place, opposite the pavilion, where he couldcatch each ray of sun—he was obsessively heliotropic.In order to deceive the sun into shining, he broughtwith him, even on a �ne May afternoon, what he calledhis ‘anti-God battery’. This consisted of three or foursweaters, an umbrella belonging to his sister, and alarge envelope containing mathematical manuscripts,such as a PhD dissertation, a paper which he was refer-eeing for the Royal Society, or some tripos answers. Hewould explain to an acquaintance that God, believingthat Hardy expected the weather to change and givehim a chance to work, counter-suggestibly arrangedthat the sky should remain cloudless.

31

There he sat. To complete his pleasure in a long af-ternoon watching cricket, he liked the sun to be shiningand a companion to join in the fun. Technique, tactics,formal beauty—those were the deepest attractions ofthe game for him. I won’t try to explain them: they areincommunicable unless one knows the language: justas some of Hardy’s classical aphorisms are inexplica-ble unless one knows either the language of cricket orof the theory of numbers, and preferably both. Fortu-nately for a good many of our friends, he also had arelish for the human comedy.

He would have been the �rst to disclaim that he hadany special psychological insight. But he was the mostintelligent of men, he had lived with his eyes open andread a lot, and he had obtained a good generalized senseof human nature—robust, indulgent, satirical, and ut-terly free from moral vanity. He was spiritually candidas few men are (I doubt if anyone could be more can-did), and he had a mocking horror of pretentiousness,self-righteous indignation, and the whole stately pan-technicon of the hypocritical virtues. Now cricket, themost beautiful of games, is also the most hypocritical.It is supposed to be the ultimate expression of the teamspirit. One ought to prefer to make o and see one’sside win, than make 100 and see it lose (one very greatplayer, like Hardy a man of innocent candor, once re-marked mildly that he never managed to feel so). Thisparticular ethos inspired Hardy’s sense of the ridicu-lous. In reply he used to expound a counterbalancingseries of maxims. Examples:

32

‘Cricket is the only game where you are playingagainst eleven of the other side and ten of your own.’

‘If you are nervous when you go in �rst, nothingrestores your con�dence so much as seeing the otherman get out.’

If his listeners were lucky, they would hear otherremarks, not relevant to cricket, as sharp-edged in con-versation as in his writing. In the Apology there aresome typical specimens: here are a few more.

‘It is never worth a �rst class man’s time to expressa majority opinion. By de�nition, there are plenty ofothers to do that.’

‘When I was an undergraduate one might, if onewere su�ciently unorthodox, suggest that Tolstoi camewithin touching distance of George Meredith as a nov-elist: but, of course, no one else possibly could.’ (Thiswas said about the intoxications of fashion: it is worthremembering that he had lived in one of the most bril-liant of all Cambridge generations.)

‘For any serious purpose, intelligence is a very mi-nor gift.’

‘Young men ought to be conceited: but they oughtn’tto be imbecile.’ (Said after someone had tried to per-suade him that Finnegans Wake was the �nal literarymasterpiece.)

‘Sometimes one has to say di�cult things, but oneought to say them as simply as one knows how.’

Occasionally, as he watched cricket, his ball-by-ballinterest �agged. Then he demanded that we shouldpick teams: teams of humbugs, club-men, bogus poets,bores, teams whose names began with HA (numbers

33

one and two Hadrian and Hannibal), SN, all-time teamsof Trinity, Christ’s, and so on. In these exercises I wasat a disadvantage: let anyone try to produce a team ofworld-�gures whose names start with SN. The Trinityteam is overwhelmingly strong (Clerk Maxwell, Byron,Thackeray, Tennyson aren’t certain of places): whileChrist’s, beginning strongly with Milton and Darwin,has nothing much to show from number 3 down.

Or he had another favorite entertainment. ‘Markthat man we met last night’, he said, and someone hadto be marked out of 100 in each of the categories Hardyhad long since invented and de�ned. Stark, Bleak (‘astark man is not necessarily bleak: but all bleak menwithout exception want to be considered stark’), Dim,Old Brandy, Spin, and some others. Stark, Bleakand Dim are self-explanatory (the Duke of Wellingtonwould get a �at 100 for Stark andBleak, and o forDim).Old Brandy was derived from a mythical characterwho said that he never drank anything but old brandy.Hence, by extrapolation, Old Brandy came to meana taste that was eccentric, esoteric, but just within thebounds of reason. As a character (and in Hardy’s view,though not mine, as a writer also) Proust got high marksfor Old Brandy: so did F.A. Lindemann (later LordCherwell).

The summer days passed. After one of the shortCambridge seasons, there was the University match.Arranging to meet him in London was not always sim-ple, for, as I mentioned before, he had a morbid suspi-cion of mechanical gadgets (he never used a watch),in particular of the telephone. In his rooms in Trinity

34

or his �at in St George’s Square, he used to say, in adisapproving and slightly sinister tone: ‘If you fancyyourself at the telephone, there is one in the next room.’Once in an emergency he had to ring me up: angrilyhis voice came at me: ‘I shan’t hear a word you say, sowhen I’m �nished I shall immediately put the receiverdown. It’s important you should come round betweennine and ten tonight.’ Clonk.

Yet, punctually, he arrived at the University match.There he was at his most sparkling, year after year.Surrounded by friends, men and women, he was quitereleased from shyness. He was the center of all ourattention, which he didn’t dislike. Sometimes one couldhear the party’s laughter from a quarter of the wayround the ground.

In those last of his happy years, everything he didwas light with grace, order, a sense of style. Cricketis a game of grace and order, which is why he foundformal beauty in it. His mathematics, so I am told,had these same aesthetic qualities, right up to his lastcreative work. I have given the impression, I fancy,that in private he was a conversational performer. Toan extent, that was true: but he was also, on whathe would have called ‘non-trivial’ occasions (meaningoccasions important to either participant) a serious andconcentrated listener. Of other eminent persons whom,by various chances. I knew at the same period, Wellswas, on the whole, a worse listener than one expected:Rutherford distinctly better: Lloyd George one of thebest listeners of all time. Hardy didn’t suck impressionsand knowledge out of others’ words, as Lloyd George

35

did, but his mind was at one’s disposal. When, yearsbefore I wrote it, he heard of the concept of The Masters,he cross-examined me, so that I did most of the talking.He produced some good ideas. I wish he had been ableto read the book, which I think that he might have liked.Anyway, in that hope I dedicated it to his memory.

In the note at the end of the Apology he refers toother discussions. One of these was long-drawn-out,and sometimes, on both sides, angry. On the secondworld war we each had passionate but, as I shall have tosay a little later, di�erent opinions. I didn’t shift his byone inch. Nevertheless, though we were separated bya gulf of emotion, on the plane of reason he recognizedwhat I was saying. That was true in any argument Ihad with him.

Through the thirties he lived his own version ofa young man’s life. Then suddenly it broke. In 1939he had a coronary thrombosis. He recovered, but realtennis, squash, the physical activities he loved, wereover for good. The war darkened him still further, justas the �rst war had. To him they were connected piecesof lunacy, we were all at fault, he couldn’t identifyhimself with the war— once it was clear the countrywould survive—any more than he had done in 1914.One of his closest friends died tragically. And—I thinkthere is no doubt these griefs were inter-connected—his creative powers as a mathematician at last, in hissixties, left him.

That is why A Mathematician’s Apology is, if readwith the textual attention it deserves, a book of haunt-ing sadness. Yes, it is witty and sharp with intellectual

36

high spirits: yes, the crystalline clarity and candor arestill there: yes, it is the testament of a creative artist.But it is also, in an understated stoical fashion, a pas-sionate lament for creative powers that used to be andthat will never come again. I know nothing like it in thelanguage: partly because most people with the literarygift to express such a lament don’t come to feel it: itis very rare for a writer to realize, with the �nality oftruth, that he is absolutely �nished.

Seeing him in those years, I couldn’t help thinkingof the price he was paying for his young man’s life. Itwas like seeing a great athlete, for years in the pride ofhis youth and skill, so much younger and more joyfulthan the rest of us, suddenly have to accept that thegift has gone. It is common to meet great athletes whohave gone, as they call it, over the hill: fairly quickly thefeet get heavier (often the eyes last longer), the strokeswon’t come o�, Wimbledon is a place to be dreaded, thecrowds go to watch someone else. That is the point atwhich a good many athletes take to drink. Hardy didn’ttake to drink: but he took to something like despair.He recovered enough physically to have ten minutesbatting at the nets, or to play his pleasing elaboration(with a complicated set of bisques) of Trinity bowls. Butit was often hard to rouse his interest—three or fouryears before his interest in everything was so sparklingas sometimes to tire us all out. ‘No one should everbe bored’, had been one of his axioms. ‘One can behorri�ed, or disgusted, but one can’t be bored.’ Yet nowhe was often just that, plain bored.

37

It was for that reason that some of his friends,including me, encouraged him to write the story ofBertrand Russell and Trinity in the 1914-18 war. Peo-ple who didn’t know how depressed Hardy was thoughtthe whole episode was now long over and ought not tobe resurrected. The truth was, it enlivened him to haveany kind of purpose. The book was privately circulated.It has never been obtainable by the public, which is apity, for it is a small-scale addition to academic history.

I used such persuasion as I had to get him to writeanother book, which he had in happier days promisedme to do. It was to be called A Day at the Oval andwas to consist of himself watching cricket for a wholeday, spreading himself in disquisitions on the game,human nature, his reminiscences, life in general. Itwould have been an eccentric minor classic: but it wasnever written.

I wasn’t much help to him in those last years. I wasdeeply involved in war-time Whitehall, I was preoccu-pied and often tired, it was an e�ort to get to Cambridge.But I ought to have made the e�ort more often than Idid. I have to admit, with remorse, that there was, notexactly a chill, but a gap in sympathy between us. Helent me his �at in Pimlico—a dark and seedy �at withthe St George’s Square gardens outside and what hecalled an ‘old brandy’ attractiveness—for the whole ofthe war. But he didn’t like me being so totally commit-ted. People he approved of oughtn’t to give themselveswhole-heartedly to military functions. He never askedme about my work. He didn’t want to talk about thewar. While I, for my part, was impatient and didn’t

38

show anything like enough consideration. After all, Ithought, I wasn’t doing this job for fun: as I had to doit, I might as well extract the maximum interest. Butthat is no excuse.

At the end of the war I did not return to Cambridge.I visited him several times in 1946. His depression hadnot lifted, he was physically failing, short of breathafter a few yards walk. The long cheerful stroll acrossParker’s Piece, after the close of play, was gone for ever:I had to take him home to Trinity in a taxi. He wasglad that I had gone back to writing books: the creativelife was the only one for a serious man. As for himself,he wished that he could live the creative life again, nobetter than it had been before: his own life was over.

I am not quoting his exact words. This was sounlike him that I wanted to forget, and I tried, by akind of irony, to smear over what had just been said.So that I have never remembered precisely. I attemptedto dismiss it to myself as a rhetorical �ourish.

In the early summer of 1947 I was sitting at break-fast when the telephone rang. It was Hardy’s sister: hewas seriously ill, would I come up to Cambridge at once,would I call at Trinity �rst? At the time I didn’t graspthe meaning of the second request. But I obeyed it,and in the porter’s lodge at Trinity that morning founda note from her: I was to go to Donald Robertson’srooms, he would be waiting for me.

Donald Robertson was the Professor of Greek, andan intimate friend of Hardy’s: he was another mem-ber of the same high, liberal, graceful Edwardian Cam-bridge. Incidentally, he was one of the few people who

39

called Hardy by his Christian name. He greeted mequietly. Outside the windows of his room it was a calmand sunny morning. He said:

‘You ought to know that Harold has tried to killhimself.’

Yes, he was out of danger: he was for the time beingall right, if that was the phrase to use. But Donald was,in a less pointed fashion, as direct as Hardy himself.It was a pity the attempt had failed. Hardy’s healthhad got worse: he could not in any case live long: evenwalking from his rooms to hall had become a strain.He had made a completely deliberate choice. Life onthose terms he would not endure: there was nothing init. He had collected enough barbiturates: he had triedto do a thorough job, and had taken too many.

I was fond of Donald Robertson, but I had met himonly at parties and at Trinity high table. This was the�rst occasion on which we had talked intimately. Hesaid, with gentle �rmness, that I ought to come up tosee Hardy as often as I could: it would be hard to take,but it was an obligation: probably it would not be forlong. We were both wretched. I said goodbye, andnever saw him again.

In the Evelyn nursing home, Hardy was lying in bed.As a touch of farce, he had a black eye. Vomiting fromthe drugs, he had hit his head on the lavatory basin.He was self-mocking. He had made a mess of it. Hadanyone ever made a bigger mess? I had to enter intothe sarcastic game. I had never felt less like sarcasm,but I had to play. I talked about other distinguishedfailures at bringing o� suicide. What about German

40

generals in the last war? Beck, Stulpnagel, they hadbeen remarkably incompetent at it. It was bizarre tohear myself saying these things. Curiously enough, itseemed to cheer him up.

After that, I went to Cambridge at least once a week.I dreaded each visit, but early on he said that he lookedforward to seeing me. He talked a little, nearly everytime I saw him, about death. He wanted it: he didn’tfear it: what was there to fear in nothingness? His hardintellectual stoicism had come back. He would not tryto kill himself again. He wasn’t good at it. He wasprepared to wait. With an inconsistency which mighthave pained him—for he, like most of his circle, believedin the rational to an extent that I thought irrational—he showed an intense hypochondriac curiosity abouthis own symptoms. Constantly he was studying theoedema of his ankles: was it greater or less that day?

Mostly, though—about �fty-�ve minutes in eachhour I was with him—I had to talk cricket. It was hisonly solace. I had to pretend a devotion to the gamewhich I no longer felt, which in fact had been lukewarmin the thirties except for the pleasure of his company.Now I had to study the cricket scores as intently aswhen I was a schoolboy. He couldn’t read for himself,but he would have known if I was blu�ng. Sometimes,for a few minutes, his old vivacity would light up. Butif I couldn’t think of another question or piece of news,he would lie there, in the kind of dark loneliness thatcomes to some people before they die.

Once or twice I tried to rouse him. wouldn’t it beworth while, even if it was a risk, to go and see one

41

more cricket match together? I was now better o�than I used to be, I said. I was prepared to stand him ataxi, his old familiar means of transport, to any cricketground he liked to name. At that he brightened. He saidthat I might have a dead man on my hands. I repliedthat I was ready to cope. I thought that he might come:he knew, I knew, that his death could only be a matterof months: I wanted to see him have one afternoon ofsomething like gaiety. The next time I visited him heshook his head in sadness and anger. No, he couldn’teven try: there was no point in trying.

It was hard enough for me to have to talk cricket. Itwas harder for his sister, a charming intelligent womanwho had never married and who had spent much of herlife looking after him. With a humorous skill not unlikehis own old form, she collected every scrap of cricketnews she could �nd, though she had never learnedanything about the game.

Once or twice the sarcastic love of the human com-edy came bursting out. Two or three weeks before hisdeath, he heard from the Royal Society that he was tobe given their highest honor, the Copley Medal. Hegave his Mephistophelian grin, the �rst time I had seenit in full splendor in all those months. ‘Now I knowthat I must be pretty near the end. When people hurryup to give you honori�c things there is exactly oneconclusion to be drawn.’

After I heard that, I think I visited him twice. Thelast time was four or �ve days before he died. Therewas an Indian test team playing in Australia, and wetalked about them.

42

It was in that same week that he told his sister: ‘IfI knew that I was going to die today, I think I shouldstill want to hear the cricket scores.’

He managed something very similar. Each eveningthat week before she left him, she read a chapter froma history of Cambridge university cricket. One suchchapter contained the last words he heard, for he diedsuddenly, in the early morning.

43

preface

I am indebted for many valuable criticisms to ProfessorC.D. Broad and Dr C.P. Snow, each of whom read myoriginal manuscript. I have incorporated the substanceof nearly all of their suggestions in my text, and haveso removed a good many crudities and obscurities.

In one case I have dealt with them di�erently. My§28 is based on a short article which I contributed toEureka (the journal of the Cambridge ArchimedeanSociety) early in the year, and I found it impossible toremodel what I had written so recently and with somuch care. Also, if I had tried to meet such importantcriticisms seriously, I should have had to expand thissection so much as to destroy the whole balance of myessay. I have therefore left it unaltered, but have addeda short statement of the chief points made by my criticsin a note at the end.

G.H.H.18 July 1940

1

It is a melancholy experience for a professional math-ematician to �nd himself writing about mathematics.The function of a mathematician is to do something, toprove new theorems, to add to mathematics, and notto talk about what he or other mathematicians havedone. Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicianshave usually similar feelings; there is no scorn moreprofound, or on the whole more justi�able, than thatof the men who make for the men who explain. Expo-sition, criticism, appreciation, is work for second-rateminds.

I can remember arguing this point once in one ofthe few serious conversations that I ever had with Hous-man. Housman, in his Leslie Stephen lecture The Nameand Nature of Poetry, had denied very emphatically thathe was a ‘critic’; but he had denied it in what seemedto me a singularly perverse way, and had expressedan admiration for literary criticism which startled andscandalized me. He had begun with a quotation fromhis inaugural lecture, delivered twenty-two years be-fore:

Whether the faculty of literary criticismis the best gift that Heaven has in its trea-suries, I cannot say; but Heaven seems tothink so, for assuredly it is the gift mostcharily bestowed. Orators and poets. . . , ifrare in comparison with blackberries, are

45

commoner than returns of Halley’s comet:literary critics are less common. . . .

And he had continued:

In these twenty-two years I have improvedin some respects and deteriorated in oth-ers, but I have not so much improved as tobecome a literary critic, nor so much de-teriorated as to fancy that I have becomeone.

It had seemed to me deplorable that a great scholar anda �ne poet should write like this, and, �nding myselfnext to him in Hall a few weeks later, I plunged in andsaid so. Did he really mean what he had said to be takenvery seriously? Would the life of the best of critics reallyhave seemed to him comparable with that of a scholarand a poet? We argued these questions all throughdinner, and I think that �nally he agreed with me. Imust not seem to claim a dialectical triumph over aman who can no longer contradict me; but ‘Perhaps notentirely’ was, in the end, his reply to the �rst question,and ‘Probably no’ to the second.

There may have been some doubt about Housman’sfeelings, and I do not wish to claim him as on my side;but there is no doubt at all about the feelings of menof science, and I share them fully. If then I �nd myselfwriting, not mathematics but ‘about’ mathematics, itis a confession of weakness, for which I may rightlybe scorned or pitied by younger and more vigorousmathematicians. I write about mathematics because,

46

like any other mathematician who has passed sixty, Ihave no longer the freshness of mind, the energy, orthe patience to carry on e�ectively with my proper job.

2

I propose to put forward an apology for mathematics;and I may be told that it needs none, since there arenow few studies more generally recognized, for goodreasons or bad, as pro�table and praiseworthy. Thismay be true; indeed it is probable, since the sensationaltriumphs of Einstein, that stellar astronomy and atomicphysics are the only sciences which stand higher inpopular estimation. A mathematician need not nowconsider himself on the defensive. He does not haveto meet the sort of opposition described by Bradley inthe admirable defense of metaphysics which forms theintroduction to Appearance and Reality.

A metaphysician, says Bradley, will be told that‘metaphysical knowledge is wholly impossible’, or that‘even if possible to a certain degree, it is practically noknowledge worth the name’. ‘The same problems,’ hewill hear, ‘the same disputes, the same sheer failure.Why not abandon it and come out? Is there nothingelse more worth your labor?’ There is no one so stupidas to use this sort of language about mathematics. Themass of mathematical truth is obvious and imposing; itspractical applications, the bridges and steam-enginesand dynamos, obtrude themselves on the dullest imagi-

47

nation. The public does not need to be convinced thatthere is something in mathematics.

All this is in its way very comforting to mathemati-cians, but it is hardly possible for a genuine mathemati-cian to be content with it. Any genuine mathematicianmust feel that it is not on these crude achievements thatthe real case for mathematics rests, that the popularreputation of mathematics is based largely on igno-rance and confusion, and that there is room for a morerational defense. At any rate, I am disposed to try tomake one. It should be a simpler task than Bradley’sdi�cult apology.

I shall ask, then, why is it really worth while tomake a serious study of mathematics? What is theproper justi�cation of a mathematician’s life? And myanswers will be, for the most part, such as are to beexpected from a mathematician: I think that it is worthwhile, that there is ample justi�cation. But I shouldsay at once that my defense of mathematics will be adefense of myself, and that my apology is bound to beto some extent egotistical. I should not think it worthwhile to apologize for my subject if I regarded myselfas one of its failures.

Some egotism of this sort is inevitable, and I do notfeel that it really needs justi�cation. Good work is notdone by ‘humble’ men. It is one of the �rst duties of aprofessor, for example, in any subject, to exaggerate alittle both the importance of his subject and his ownimportance in it. A man who is always asking ‘Is whatI do worth while?’ and ‘Am I the right person to do it?’will always be ine�ective himself and a discouragement

48

to others. He must shut his eyes a little and think alittle more of his subject and himself than they deserve.This is not too di�cult: it is harder not to make hissubject and himself ridiculous by shutting his eyes tootightly.

3

A man who sets out to justify his existence and hisactivities has to distinguish two di�erent questions.The �rst is whether the work which he does is worthdoing; and the second is why he does it, whatever itsvalue may be. The �rst question is often very di�cult,and the answer very discouraging, but most people will�nd the second easy enough even then. Their answers,if they are honest, will usually take one or other oftwo forms; and the second form is merely a humblervariation of the �rst, which is the only answer whichwe need consider seriously.

(1) ‘I do what I do because it is the one and onlything that I can do at all well. I am a lawyer, or a stock-broker, or a professional cricketer, because I have somereal talent for that particular job. I am a lawyer be-cause I have a �uent tongue, and am interested in legalsubtleties; I am a stockbroker because my judgmentof the markets is quick and sound; I am a professionalcricketer because. I can bat unusually well. I agree thatit might be better to be a poet or a mathematician, butunfortunately I have no talent for such pursuits.’

49

I am not suggesting that this is a defense whichcan be made by most people, since most people can donothing at all well. But it is impregnable when it canbe made without absurdity, as it can by a substantialminority: perhaps �ve or even ten per cent of men cando something rather well. It is a tiny minority who cando anything really well, and the number of men whocan do two things well is negligible. If a man has anygenuine talent, he should be ready to make almost anysacri�ce in order to cultivate it to the full.

This view was endorsed by Dr Johnson:

When I told him that I had been to see[his namesake] Johnson ride upon threehorses, he said ‘Such a man, sir, shouldbe encouraged, for his performances showthe extent of the human powers’

and similarly he would have applauded mountain climbers,channel swimmers, and blindfold chess-players. Formy own part, I am entirely in sympathy with all suchattempts at remarkable achievement. I feel some sympa-thy even with conjurors and ventriloquists; and whenAlekhine and Bradman set out to beat records, I amquite bitterly disappointed if they fail. And here bothDr Johnson and I �nd ourselves in agreement withthe public. As W.J. Turner has said so truly, it is onlythe ‘highbrows’ (in the unpleasant sense) who do notadmire the ‘real swells’.

We have of course to take account of the di�erencesin value between di�erent activities. I would rather bea novelist or a painter than a statesman of similar rank;

50

and there are many roads to fame which most of uswould reject as actively pernicious. Yet it is seldom thatsuch di�erences of value will turn the scale in a man’schoice of a career, which will almost always be dictatedby the limitations of his natural abilities. Poetry is morevaluable than cricket, but Bradman would be a fool ifhe sacri�ced his cricket in order to write second-rateminor poetry (and I suppose that it is unlikely that hecould do better). If the cricket were a little less supreme,and the poetry better, then the choice might be moredi�cult: I do not know whether I would rather havebeen Victor Trumper or Rupert Brooke. It is fortunatethat such dilemmas occur so seldom.

I may add that they are particularly unlikely topresent themselves to a mathematician. It is usual toexaggerate rather grossly the di�erences between themental processes of mathematicians and other people,but it is undeniable that a gift for mathematics is one ofthe most specialized talents, and that mathematiciansas a class are not particularly distinguished for generalability or versatility. If a man is in any sense a realmathematician, then it is a hundred to one that hismathematics will be far better than anything else hecan do, and that he would be silly if he surrenderedany decent opportunity of exercising his one talent inorder to do undistinguished work in other �elds. Sucha sacri�ce could be justi�ed only by economic necessityor age.

51

4

I had better say something here about this question ofage, since it is particularly important for mathemati-cians. No mathematician should ever allow himselfto forget that mathematics, more than any other artor science, is a young man’s game. To take a simpleillustration at a comparatively humble level, the aver-age age of election to the Royal Society is lowest inmathematics.

We can naturally �nd much more striking illustra-tions. We may consider, for example, the career of aman who was certainly one of the world’s three great-est mathematicians. Newton gave up mathematics at�fty, and had lost his enthusiasm long before; he hadrecognized no doubt by the time that he was forty thathis great creative days were over. His greatest ideasof all, �uxions and the law of gravitation, came to himabout 1666, when he was twenty-four—‘in those days Iwas in the prime of my age for invention, and mindedmathematics and philosophy more than at any timesince’. He made big discoveries until he was nearlyforty (the ‘elliptic orbit’ at thirty-seven), but after thathe did little but polish and perfect.

Galois died at twenty-one, Abel at twenty-seven,Ramanujan at thirty-three, Riemann at forty. Therehave been men who have done great work a good deallater; Gauss’s great memoir on di�erential geometrywas published when he was �fty (though he had hadthe fundamental ideas ten years before). I do not know

52

an instance of a major mathematical advance initiatedby a man past �fty. If a man of mature age loses interestin and abandons mathematics, the loss is not likely tobe very serious either for mathematics or for himself.

On the other hand the gain is no more likely tobe substantial; the later records of mathematicianswho have left mathematics are not particularly en-couraging. Newton made a quite competent Master ofthe Mint (when he was not quarreling with anybody).Painleve was a not very successful Premier of France.Laplace’s political career was highly discreditable, buthe is hardly a fair instance, since he was dishonestrather than incompetent, and never really ‘gave up’mathematics. It is very hard to �nd an instance of a�rst-rate mathematician who has abandoned mathe-matics and attained �rst-rate distinction in any other�eld.1 There may have been young men who wouldhave been �rst-rate mathematicians if they had stuckto mathematics, but I have never heard of a really plau-sible example. And all this is fully borne out by my ownvery limited experience. Every young mathematicianof real talent whom I have known has been faithful tomathematics, and not from lack of ambition but fromabundance of it; they have all recognized that there, ifanywhere, lay the road to a life of any distinction.

1Pascal seems the best.

53

5

There is also what I called the ‘humbler variation’ ofthe standard apology; but I may dismiss this in a veryfew words.

(2) ‘There is nothing that I can do particularly well.I do what I do because it came my way. I really neverhad a chance of doing anything else.’ And this apologytoo I accept as conclusive. It is quite true that mostpeople can do nothing well. If so, it matters very littlewhat career they choose, and there is really nothingmore to say about it. It is a conclusive reply, but hardlyone likely to be made by a man with any pride; and Imay assume that none of us would be content with it.

6

It is time to begin thinking about the �rst questionwhich I put in §3, and which is so much more di�cultthan the second. Is mathematics, what I and othermathematicians mean by mathematics, worth doing;and if so, why?

I have been looking again at the �rst pages of theinaugural lecture which I gave at Oxford in 1920, wherethere is an outline of an apology for mathematics. Itis very inadequate (less than a couple of pages), andit is written in a style (a �rst essay, I suppose, in whatI then imagined to be the ‘Oxford manner’) of whichI am not now particularly proud; but I still feel that,however much development it may need, it contains

54

the essentials of the matter. I will resume what I saidthen, as a preface to a fuller discussion.

(1) I began by laying stress on the harmlessnessof mathematics—‘the study of mathematics is, if anunpro�table, a perfectly harmless and innocent occu-pation’. I shall stick to that, but obviously it will needa good deal of expansion and explanation.

Is mathematics ‘unpro�table’? In some ways, plainly,it is not; for example, it gives great pleasure to quitea large number of people. I was thinking of ‘pro�t’,however, in a narrower sense. Is mathematics ‘use-ful’, directly useful, as other sciences such as chemistryand physiology are? This is not an altogether easyor uncontroversial question, and I shall ultimately sayNo, though some mathematicians, and most outsiders,would no doubt say Yes. And is mathematics ‘harm-less’? Again the answer is not obvious, and the questionis one which I should have in some ways preferred toavoid, since it raises the whole problem of the e�ect ofscience on war. Is mathematics harmless, in the sensein which, for example, chemistry plainly is not? I shallhave to come back to both these questions later.

(2) I went on to say that ‘the scale of the universe islarge and, if we are wasting our time, the waste of thelives of a few university dons is no such overwhelmingcatastrophe’: and here I may seem to be adopting, ora�ecting, the pose of exaggerated humility which Irepudiated a moment ago. I am sure that that was notwhat was really in my mind; I was trying to say in asentence what I have said at much greater length in§3. I was assuming that we dons really had our little

55

talents, and that we could hardly be wrong if we didour best to cultivate them fully.

(3) Finally (in what seem to me now some ratherpainfully rhetorical sentences) I emphasized the per-manence of mathematical achievement:

What we do may be small, but it has acertain character of permanence; and tohave produced anything of the slightestpermanent interest, whether it be a copyof verses or a geometrical theorem, is tohave done something utterly beyond thepowers of the vast majority of men.

And:

In these days of con�ict between ancientand modern studies, there must surely besomething to be said for a study which didnot begin with Pythagoras, and will notend with Einstein, but is the oldest and theyoungest of all.

All this is ‘rhetoric’; but the substance of it seems tome still to ring true, and I can expand it at once with-out prejudging any of the other questions which I amleaving open.

7

I shall assume that I am writing for readers who arefull, or have in the past been full, of a proper spirit of

56

ambition. A man’s �rst duty, a young man’s at anyrate, is to be ambitious. Ambition is a noble passionwhich may legitimately take many forms; there wassomething noble in the ambition of Attila or Napoleon:but the noblest ambition is that of leaving behind onesomething of permanent value:

Here, on the level sand,Between the sea and land,What shall I build or writeAgainst the fall of night?

Tell me of runes to graveThat hold the bursting wave,Or bastions to designFor longer date than mine.

Ambition has been the driving force behind nearly allthe best work of the world. In particular, practicallyall substantial contributions to human happiness havebeen made by ambitious men. To take two famous ex-amples, were not Lister and Pasteur ambitious? Or, ona humbler level, King Gillette and William Willett; andwho in recent times have contributed more to humancomfort than they?

Physiology provides particularly good examples,just because it is so obviously a ‘bene�cial’ study. Wemust guard against a fallacy common among apologistsof science, the fallacy of supposing that the men whosework most bene�ts humanity are thinking much of thatwhile they do it, that physiologists, for example, haveparticularly noble souls. A physiologist may indeed be

57

glad to remember that his work will bene�t mankind,but the motives which provide the force and the in-spiration for it are indistinguishable from those of aclassical scholar or a mathematician.

There are many highly respectable motives whichmay lead men to prosecute research, but three whichare much more important than the rest. The �rst (with-out which the rest must come to nothing) is intellectualcuriosity, desire to know the truth. Then, professionalpride, anxiety to be satis�ed with one’s performance,the shame that overcomes any self-respecting crafts-man when his work is unworthy of his talent. Finally,ambition, desire for reputation, and the position, eventhe power or the money, which it brings. It may be�ne to feel, when you have done your work, that youhave added to the happiness or alleviated the su�eringsof others, but that will not be why you did it. So if amathematician, or a chemist, or even a physiologist,were to tell me that the driving force in his work hadbeen the desire to bene�t humanity, then I should notbelieve him (nor should I think the better of him if Idid). His dominant motives have been those which Ihave stated, and in which, surely, there is nothing ofwhich any decent man need be ashamed.

8

If intellectual curiosity, professional pride, and am-bition are the dominant incentives to research, thenassuredly no one has a fairer chance of gratifying them

58

than a mathematician. His subject is the most curious ofall—there is none in which truth plays such odd pranks.It has the most elaborate and the most fascinating tech-nique, and gives unrivaled openings for the displayof sheer professional skill. Finally, as history provesabundantly, mathematical achievement, whatever itsintrinsic worth, is the most enduring of all.

We can see this even in semi-historic civilizations.The Babylonian and Assyrian civilizations have per-ished; Hammurabi, Sargon, and Nebuchadnezzar areempty names; yet Babylonian mathematics is still in-teresting, and the Babylonian scale of 60 is still used inastronomy. But of course the crucial case is that of theGreeks.

The Greeks were the �rst mathematicians who arestill ‘real’ to us to-day. Oriental mathematics may be aninteresting curiosity, but Greek mathematics is the realthing. The Greeks �rst spoke a language which modernmathematicians can understand; as Littlewood said tome once, they are not clever schoolboys or ‘scholar-ship candidates’, but ‘Fellows of another college’. SoGreek mathematics is ‘permanent’, more permanenteven than Greek literature. Archimedes will be remem-bered when Aeschylus is forgotten, because languagesdie and mathematical ideas do not. ‘Immortality’ maybe a silly word, but probably a mathematician has thebest chance of whatever it may mean.

Nor need he fear very seriously that the future willbe unjust to him. Immortality is often ridiculous orcruel: few of us would have chosen to be Og or Ana-nias or Gallio. Even in mathematics, history sometimes

59

plays strange tricks; Rolle �gures in the text-books ofelementary calculus as if he had been a mathematicianlike Newton; Farey is immortal because he- failed tounderstand a theorem which Haros had proved per-fectly fourteen years before; the names of �ve worthyNorwegians still stand in Abel’s Life, just for one actof conscientious imbecility, dutifully per-formed at theexpense of their country’s greatest man. But on thewhole the history of science is fair, and this is particu-larly true in mathematics. No other subject has suchclear cut or unanimously accepted standards, and themen who are remembered are almost always the menwho merit it. Mathematical fame, if you have the cashto pay for it, is one of the soundest and steadiest ofinvestments.

9

All this is very comforting for dons, and especiallyfor professors of mathematics. It is sometimes sug-gested, by lawyers or politicians or business men, thatan academic career is one sought mainly by cautiousand unambitious persons who care primarily for com-fort and security. The reproach is quite misplaced. Adon surrenders something, and in particular the chanceof making large sums of money—it is very hard for aprofessor to make £2000 a year; and security of tenureis naturally one of the considerations which make thisparticular surrender easy. That is not why Housmanwould have refused to be Lord Simon or Lord Beaver-

60

brook. He would have rejected their careers because ofhis ambition, because he would have scorned to be aman to be forgotten in twenty years.

Yet how painful it is to feel that, with all theseadvantages, one may fail. I can remember BertrandRussell telling me of a horrible dream. He was in thetop �oor of the University Library, about a.d. 2100. A li-brary assistant was going round the shelves carrying anenormous bucket, taking down book after book, glanc-ing at them, restoring them to the shelves or dumpingthem into the bucket. At last he came to three largevolumes which Russell could recognize as the last sur-viving copy of Principia Mathematica. He took downone of the volumes, turned over a few pages, seemedpuzzled for a moment by the curious symbolism, closedthe volume, balanced it in his hand and hesitated. . . .

10

A mathematician, like a painter or a poet, is a makerof patterns. If his patterns are more permanent thantheirs, it is because they are made with ideas. A paintermakes patterns with shapes and colors, a poet withwords. A painting may embody an ‘idea’, but the ideais usually commonplace and unimportant. In poetry,ideas count for a good deal more; but, as Housmaninsisted, the importance of ideas in poetry is habituallyexaggerated: ‘I cannot satisfy myself that there are anysuch things as poetical ideas. . . .

Poetry is not the thing said but a way of saying it.’

61

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 moretrite and more false? The poverty of the ideas seemshardly to a�ect the beauty of the verbal pattern. Amathematician, on the other hand, has no material towork with but ideas, and so his patterns are likely tolast longer, since ideas wear less with time than words.

The mathematician’s patterns, like the painter’s orthe poet’s, must be beautiful; the ideas, like the colorsor the words, must �t together in a harmonious way.Beauty is the �rst test: there is no permanent placein the world for ugly mathematics. And here I mustdeal with a misconception which is still widespread(though probably much less so now than it was twentyyears ago), what Whitehead has called the ‘literary su-perstition’ that love of and aesthetic appreciation ofmathematics is ‘a monomania con�ned to a few ec-centrics in each generation’.

It would be di�cult now to �nd an educated manquite insensitive to the aesthetic appeal of mathematics.It may be very hard to de�ne mathematical beauty, butthat is just as true of beauty of any kind—we may notknow quite what we mean by a beautiful poem, but thatdoes not prevent us from recognizing one when we readit. Even Professor Hogben, who is out to minimize at allcosts the importance of the aesthetic element in mathe-matics, does not venture to deny its reality. ‘There are,to be sure, individuals for whom mathematics exercisesa coldly impersonal attraction The aesthetic appeal of

62

mathematics may be very real for a chosen few.’ Butthey are ‘few’, he suggests, and they feel ‘coldly’ (andare really rather ridiculous people, who live in sillylittle university towns sheltered from the fresh breezesof the wide open spaces). In this he is merely echoingWhitehead’s ‘literary superstition’.

The fact is that there are few more ‘popular’ sub-jects than mathematics. Most people have some appre-ciation of mathematics, just as most people can enjoya pleasant tune; and there are probably more peoplereally interested in mathematics than in music. Ap-pearances may suggest the contrary, but there are easyexplanations. Music can be used to stimulate mass emo-tion, while mathematics cannot; and musical incapacityis recognized (no doubt rightly) as mildly discreditable,whereas most people are so frightened of the name ofmathematics that they are ready, quite una�ectedly, toexaggerate their own mathematical stupidity.

A very little re�ection is enough to expose the ab-surdity of the ‘literary superstition’. There are massesof chess-players in every civilized country—in Russia,almost the whole educated population; and every chess-player can recognize and appreciate a ‘beautiful’ gameor problem. Yet a chess problem is simply an exercisein pure mathematics (a game not entirely, since psy-chology also plays a part), and everyone who calls aproblem ‘beautiful’ is applauding mathematical beauty,even if it is beauty of a comparatively lowly kind. Chessproblems are the hymn-tunes of mathematics.

We may learn the same lesson, at a lower level butfor a wider public, from bridge, or descending further,

63

from the puzzle columns of the popular newspapers.Nearly all their immense popularity is a tribute to thedrawing power of rudimentary mathematics, and thebetter makers of puzzles, such as Dudeney or ‘Caliban’,use very little else. They know their business; what thepublic wants is a little intellectual ‘kick’, and nothingelse has quite the kick of mathematics. I might add thatthere is nothing in the world which pleases even famousmen (and men who have used disparaging languageabout mathematics) quite so much as to discover, orrediscover, a genuine mathematical theorem. HerbertSpencer republished in his autobiography a theoremabout circles which he proved when he was twenty (notknowing that it had been proved over two thousandyears before by Plato). Professor Soddy is a more recentand a more striking example (but his theorem really ishis own)2.

11

A chess problem is genuine mathematics, but it is insome way ‘trivial’ mathematics. However ingeniousand intricate, however original and surprising the moves,there is something essential lacking. Chess problemsare unimportant. The best mathematics is serious aswell as beautiful—‘important’ if you like, but the wordis very ambiguous, and ‘serious’ expresses what I meanmuch better.

2See his letters on the ‘Hexlet’ in Nature, vols. 137-9 (1936-7)

64

I am not thinking of the ‘practical’ consequencesof mathematics. I have to return to that point later: atpresent I will say only that if a chess problem is, in thecrude sense, ‘useless’, then that is equally true of mostof the best mathematics; that very little of mathematicsis useful practically, and that that little is comparativelydull. The ‘seriousness’ of a mathematical theorem lies,not in its practical consequences, which are usually neg-ligible, but in the signi�cance of the mathematical ideaswhich it connects. We may say, roughly, that a mathe-matical idea is ‘signi�cant’ if it can be connected, in anatural and illuminating way, with a large complex ofother mathematical ideas. Thus a serious mathematicaltheorem, a theorem which connects signi�cant ideas,is likely to lead to important advances in mathematicsitself and even in other sciences. No chess problemhas ever a�ected the general development of scienti�cthought; Pythagoras, Newton, Einstein have in theirtimes changed its whole direction.

The seriousness of a theorem, of course, does notlie in its consequences, which are merely the evidencefor its seriousness. Shakespeare had an enormous in-�uence on the development of the English language,Otway next to none, but that is not why Shakespearewas the better poet. He was the better poet becausehe wrote much better poetry. The inferiority of thechess problem, like that of Otway’s poetry, lies not inits consequences but in its content.

There is one more point which I shall dismiss veryshortly, not because it is uninteresting but because itis di�cult, and because I have no quali�cations for

65

any serious discussion in aesthetics. The beauty ofa mathematical theorem depends a great deal on itsseriousness, as even in poetry the beauty of a line maydepend to some extent on the signi�cance of the ideaswhich it contains. I quoted two lines of Shakespeare asan example of the sheer beauty of a verbal pattern; but

After life’s �tful fever he sleeps well

seems still more beautiful. The pattern is just as �ne,and in this case the ideas have signi�cance and thethesis is sound, so that our emotions are stirred muchmore deeply. The ideas do matter to the pattern, evenin poetry, and much more, naturally, in mathematics;but I must not try to argue the question seriously.

12

It will be clear by now that, if we are to have anychance of making progress, I must produce examplesof ‘real’ mathematical theorems, theorems which everymathematician will admit to be �rst-rate. And here Iam very heavily handicapped by the restrictions underwhich I am writing. On the one hand my examplesmust be very simple, and intelligible to a reader whohas no specialized mathematical knowledge; no elabo-rate preliminary explanations must be needed; and areader must be able to follow the proofs as well as theenunciations. These conditions exclude, for instance,many of the most beautiful theorems of the theory ofnumbers, such as Fermat’s ‘two square’ theorem or the

66

law of quadratic reciprocity. And on the other handmy examples should be drawn from ‘pukka’ mathe-matics, the mathematics of the working professionalmathematician; and this condition excludes a good dealwhich it would be comparatively easy to make intelli-gible but which trespasses on logic and mathematicalphilosophy.

I can hardly do better than go back to the Greeks.I will state and prove two of the famous theorems ofGreek mathematics. They are ‘simple’ theorems, simpleboth in idea and in execution, but there is no doubt at allabout their being theorems of the highest class. Each isas fresh and signi�cant as when it was discovered—twothousand years have not written a wrinkle on eitherof them. Finally, both the statements and the proofscan be mastered in an hour by any intelligent reader,however slender his mathematical equipment.

(1) The �rst is Euclid’s proof of the existence of anin�nity of prime numbers.3

The prime numbers or primes are the numbers

2,3,5,7,11,13,17,19,23,29, . . . (A)

which cannot be resolved into smaller factors4. Thus 37and 317 are prime. The primes are the material out ofwhich all numbers are built up by multiplication: thus

3Elements ix 20. The real origin of many theorems in the El-ements is obscure, but there seems to be no particular reason forsupposing that this is not Euclid’s own.

4There are technical reasons for not counting 1 as a prime.

67

666 = 2 × 3 × 3 × 37.

Every number which is not prime itself is divisible byat least one prime (usually, of course, by several). Wehave to prove that there are in�nitely many primes,i.e. that the series (A) never comes to an end. Let ussuppose that it does, and that

2,3,5, . . . ,P

is the complete series (so that P is the largest prime);and let us, on this hypothesis, consider the number Qde�ned by the formula

Q = (2 × 3 × 5 × . . . × P ) + 1

It is plain that Q is not divisible by any of 2, 3, 5, . . . , P ;for it leaves the remainder 1 when divided by any one ofthese numbers. But, if not itself prime, it is divisible bysome prime, and therefore there is a prime (which maybe Q itself) greater than any of them. This contradictsour hypothesis, that there is no prime greater than P ;and therefore this hypothesis is false.

The proof is by reductio ad absurdum, and reductioad absurdum, which Euclid loved so much, is one of amathematician’s �nest weapons.5 It is a far �ner gambitthan any chess gambit: a chess player may o�er thesacri�ce of a pawn or even a piece, but a mathematiciano�ers the game.

5The proof can be arranged so as to avoid a reductio, and logi-cians of some schools would prefer that it should be.

68

13

2. My second example is Pythagoras’s proof of the‘irrationality’ of

√2.6

A ‘rational number’ is a fraction a/b where a andb are integers; we may suppose that a and b have nocommon factor, since if they had we could remove it.To say that ‘

√2 is irrational’ is merely another way

of saying that 2 cannot be expressed in the form(ab

)2;

and this is the same thing as saying that the equation

a2 = 2b2 (B)

cannot be satis�ed by integral values of a and b whichhave no common factor. This is a theorem of purearithmetic, which does not demand any knowledge of‘irrational numbers’ or depend on any theory abouttheir nature.

We argue again by reductio ad absurdum; we sup-pose that (B) is true, a and b being integers withoutany common factor. It follows from (B) that a2 is even(since 2b2 is divisible by 2), and therefore that a is even(since the square of an odd number is odd). If a is eventhen

a = 2c (C)

for some integral value of c; and therefore6The proof traditionally ascribed to Pythagoras, and certainly a

product of his school. The theorem occurs, in a much more generalform, in Euclid (Elements x 9).

69

2b2 = a2 = (2c )2 = 4c2

or

b2 = 2c2 (D)

Hence b2 is even, and therefore (for the same reason asbefore) b is even. That is to say, a and b are both even,and so have the common factor 2. This contradicts ourhypothesis, and therefore the hypothesis is false.

It follows from Pythagoras’s theorem that the di-agonal of a square is incommensurable with the side(that their ratio is not a rational number, that there isno unit of which both are integral multiples). For if wetake the side as our unit of length, and the length ofthe diagonal is d, then, by a very familiar theorem alsoascribed to Pythagoras7

d2 = 12 + 12 = 2,

so that d cannot be a rational number.I could quote any number of �ne theorems from

the theory of numbers whose meaning anyone canunderstand. For example, there is what is called ‘thefundamental theorem of arithmetic’, that any integercan be resolved, in one way only, into a product ofprimes. Thus

666 = 2 × 3 × 3 × 37,7Euclid, Elements i 47.

70

and there is no other decomposition; it is impossiblethat

666 = 2 × 11 × 29

or that

13 × 89 = 17 × 73

(and we can see so without working out the products).This theorem is, as its name implies, the foundation ofhigher arithmetic; but the proof, although not ‘di�cult’,requires a certain amount of preface and might be foundtedious by an unmathematical reader.

Another famous and beautiful theorem is Fermat’s‘two square’ theorem. The primes may (if we ignore thespecial prime 2) be arranged in two classes; the primes

5,13,17,29,37,41, . . .

which leave remainder 1 when divided by 4, and theprimes

3,7,11,19,23,31, . . .

which leave remainder 3. All the primes of the �rstclass, and none of the second, can be expressed as thesum of two integral squares: thus

71

5 = 12 + 22

13 = 22 + 32

17 = 12 + 42

29 = 22 + 52

but 3, 7, 11, and 19 are not expressible in this way (asthe reader may check by trial). This is Fermat’s theo-rem, which is ranked, very justly, as one of the �nest ofarithmetic. Unfortunately there is no proof within thecomprehension of anybody but a fairly expert mathe-matician.

There are also beautiful theorems in the ‘theory ofaggregates’ (Mengenlehre), such as Cantor’s theoremof the ‘non-enumerability’ of the continuum. Herethere is just the opposite di�culty. The proof is easyenough, when once the language has been mastered,but considerable explanation is necessary before themeaning of the theorem becomes clear. So I will not tryto give more examples. Those which I have given aretest cases, and a reader who cannot appreciate them isunlikely to appreciate anything in mathematics.

I said that a mathematician was a maker of pat-terns of ideas, and that beauty and seriousness werethe criteria by which his patterns should be judged. Ican hardly believe that anyone who has understood thetwo theorems will dispute that they pass these tests. Ifwe compare them with Dudeney’s most ingenious puz-zles, or the �nest chess problems that masters of thatart have composed, their superiority in both respects

72

stands out: there is an unmistakable di�erence of class.They are much more serious, and also much more beau-tiful; can we de�ne, a little more closely, where theirsuperiority lies?

14

In the �rst place, the superiority of the mathematicaltheorems in seriousness is obvious and overwhelming.The chess problem is the product of an ingenious butvery limited complex of ideas, which do not di�er fromone another very fundamentally and have no externalrepercussions. We should think in the same way ifchess had never been invented, whereas the theoremsof Euclid and Pythagoras have in�uenced thought pro-foundly, even outside mathematics.

Thus Euclid’s theorem is vital for the whole struc-ture of arithmetic. The primes are the raw materialout of which we have to build arithmetic, and Euclid’stheorem assures us that we have plenty of materialfor the task. But the theorem of Pythagoras has widerapplications and provides a better text.

We should observe �rst that Pythagoras’s argumentis capable of far-reaching extension, and can be applied,with little change of principle, to very wide classes of‘irrationals’. We can prove very similarly (as Theodorusseems to have done) that

√3,√5,√7,√11,√13,√17

are irrational, or (going beyond Theodorus) that

73

3√2, 3√17

are irrational.8Euclid’s theorem tells us that we have a good sup-

ply of material for the construction of a coherent arith-metic of the integers. Pythagoras’s theorem and itsextensions tell us that, when we have constructed thisarithmetic, it will not prove su�cient for our needs,since there will be many magnitudes which obtrudethemselves upon our attention and which it will beunable to measure; the diagonal of the square is merelythe most obvious example. The profound importanceof this discovery was recognized at once by the Greekmathematicians. They had begun by assuming (in ac-cordance, I suppose, with the ‘natural’ dictates of ‘com-mon sense’) that all magnitudes of the same kind arecommensurable, that any two lengths, for example, aremultiples of some common unit, and they had con-structed a theory of proportion based on this assump-tion. Pythagoras’s discovery exposed the unsoundnessof this foundation, and led to the construction of themuch more profound theory of Eudoxus which is setout in the �fth book of the Elements, and which is re-garded by many modern mathematicians as the �nestachievement of Greek mathematics. This theory is as-tonishingly modern in spirit, and may be regarded asthe beginning of the modern theory of irrational num-

8See Ch. iv of Hardy and Wright’s Introduction to the Theory ofNumbers, where there are discussions of di�erent generalizations ofPythagoras’s argument, and of a historical puzzle about Theodorus.

74

ber, which has revolutionized mathematical analysisand had much in�uence on recent philosophy.

There is no doubt at all, then, of the ‘seriousness’ ofeither theorem. It is therefore the better worth remark-ing that neither theorem has the slightest ‘practical’importance. In practical applications we are concernedonly with comparatively small numbers; only stellar as-tronomy and atomic physics deal with ‘large’ numbers,and they have very little more practical importance, asyet, than the most abstract pure mathematics. I do notknow what is the highest degree of accuracy which isever useful to an engineer—we shall be very generousif we say ten signi�cant �gures. Then

3.14159265

(the value of π to eight places of decimals) is the ratio

314159265100000000

of two numbers of nine digits. The number of primesless than 1,000,000,000 is 50,847,478: that is enough foran engineer, and he can be perfectly happy withoutthe rest. So much for Euclid’s theorem; and, as regardsPythagoras’s, it is obvious that irrationals are uninter-esting to an engineer, since he is concerned only withapproximations, and all approximations are rational.

75

15

A ‘serious’ theorem is a theorem which contains ‘sig-ni�cant’ ideas, and I suppose that I ought to try toanalyze a little more closely the qualities which makea mathematical idea signi�cant. This is very di�cult,and it is unlikely that any analysis which I can givewill be very valuable. We can recognize a ‘signi�cant’idea when we see it, as we can those which occur inmy two standard theorems; but this power of recog-nition requires a rather high degree of mathematicalsophistication, and of that familiarity with mathemati-cal ideas which comes only from many years spent intheir company. So I must attempt some sort of analysis;and it should be possible to make one which, howeverinadequate, is sound and intelligible so far as it goes.There are two things at any rate which seem essential,a certain generality and a certain depth; but neitherquality is easy to de�ne at all precisely.

A signi�cant mathematical idea, a serious mathe-matical theorem, should be ‘general’ in some such senseas this. The idea should be one which is a constituentin many mathematical constructs, which is used in theproof of theorems of many di�erent kinds. The the-orem should be one which, even if stated originally(like Pythagoras’s theorem) in a quite special form, iscapable of considerable extension and is typical of awhole class of theorems of its kind. The relations re-vealed by the proof should be such as connect manydi�erent mathematical ideas. All this is very vague,

76

and subject to many reservations. But it is easy enoughto see that a theorem is unlikely to be serious when itlacks these qualities conspicuously; we have only totake examples from the isolated curiosities in whicharithmetic abounds. I take two, almost at random, fromRouse Ball’s Mathematical Recreations.9

(a) 8712 and 9801 are the only four-�gure numberswhich are integral multiples of their ‘reversals’:

8712 = 4 × 21789801 = 9 × 1089

and there are no other numbers below 10,000 whichhave this property.

(b) There are just four numbers (after 1) which arethe sums of the cubes of their digits, viz.

153 = 13 + 53 + 33

370 = 33 + 73 + 03

371 = 33 + 73 + 13

407 = 43 + 03 + 73

These are odd facts, very suitable for puzzle columnsand likely to amuse amateurs, but there is nothing inthem which appeals much to a mathematician. Theproofs are neither di�cult nor interesting—merely alittle tiresome. The theorems are not serious; and it

9oldstylenums11th Edition, oldstylenums1939¿ (Revised byH.S.M. Coxeter.

77

is plain that one reason (though perhaps not the mostimportant) is the extreme specialty of both the enun-ciations and the proofs, which are not capable of anysigni�cant generalization.

16

‘Generality’ is an ambiguous and rather dangerousword, and we must be careful not to allow it to dominateour discussion too much. It is used in various sensesboth in mathematics and in writings about mathemat-ics, and there is one of these in particular, on whichlogicians have very properly laid great stress, which isentirely irrelevant here. In this sense, which is quiteeasy to de�ne, all mathematical theorems are equallyand completely ‘general’.

‘The certainty of mathematics’, says Whitehead,‘depends on its complete abstract generality.10’ Whenwe assert that 2 + 3 = 5, we are asserting a relationbetween three groups of ‘things’; and these ‘things’ arenot apples or pennies, or things of any one particularsort or another, but just things, ‘any old things’. Themeaning of the statement is entirely independent ofthe individualities of the members of the groups. Allmathematical ‘objects’ or ‘entities’ or ‘relations’, suchas ‘2’, ‘3’, ‘5’, ‘+’, or ‘=’, and all mathematical proposi-tions in which they occur, are completely general inthe sense of being completely abstract. Indeed one of

10Science and the Modern World, p.33.

78

Whitehead’s words is super�uous, since generality, inthis sense, is abstractness.

This sense of the word is important, and the logi-cians are quite right to stress it, since it embodies atruism which a good many people who ought to knowbetter are apt to forget. It is quite common, for example,for an astronomer or a physicist to claim that he hasfound a ‘mathematical proof’ that the physical universemust behave in a particular way. All such claims, ifinterpreted literally, are strictly nonsense. It cannot bepossible to prove mathematically that there will be aneclipse to-morrow, because eclipses, and other physi-cal phenomena, do not form part of the abstract worldof mathematics; and this, I suppose, all astronomerswould admit when pressed, however many eclipsesthey may have predicted correctly.

It is obvious that we are not concerned with thissort of ‘generality’ now. We are looking for di�erencesof generality between one mathematical theorem andanother, and in Whitehead’s sense all are equally gen-eral. Thus the ‘trivial’ theorems (a) and (b) of §15 arejust as ‘abstract’ or ‘general’ as those of Euclid andPythagoras, and so is a chess problem. It makes nodi�erence to a chess problem whether the pieces arewhite and black, or red and green, or whether there arephysical ‘pieces’ at all; it is the same problem which anexpert carries easily in his head and which we have toreconstruct laboriously with the aid of the board. Theboard and the pieces are mere devices to stimulate oursluggish imaginations, and are no more essential to the

79

problem than the blackboard and the chalk are to thetheorems in a mathematical lecture.

It is not this kind of generality, common to all math-ematical theorems, which we are looking for now, butthe more subtle and elusive kind of generality which Itried to describe in rough terms in §15. And we mustbe careful not to lay too much stress even on gener-ality of this kind (as I think logicians like Whiteheadtend to do). It is not mere ‘piling of subtlety of gen-eralization upon subtlety of generalization’11 which isthe outstanding achievement of modern mathematics.Some measure of generality must be present in anyhigh-class theorem, but too much tends inevitably toinsipidity. ‘Everything is what it is, and not anotherthing’, and the di�erences between things are quite asinteresting as their resemblances. We do not chooseour friends because they embody all the pleasant quali-ties of humanity, but because they are the people thatthey are. And so in mathematics; a property commonto too many objects can hardly be very exciting, andmathematical ideas also become dim unless they haveplenty of individuality. Here at any rate I can quoteWhitehead on my side: ‘it is the large generalization,limited by a happy particularity, which is the fruitfulconception.12’

11Science and the Modern World, p.44.12Science and the Modern World, p.46.

80

l7

The second quality which I demanded in a signi�cantidea was depth, and this is still more di�cult to de�ne.It has something to do with di�culty, the ‘deeper’ ideasare usually the harder to grasp: but it is not at all thesame. The ideas underlying Pythagoras’s theorem andits generalizations are quite deep, but no mathemati-cian now would �nd them di�cult. On the other hand atheorem may be essentially super�cial and yet quite dif-�cult to prove (as are many ‘Diophantine’ theorems, i.e.theorems about the solution of equations in integers).

It seems that mathematical ideas are arranged some-how in strata, the ideas in each stratum being linkedby a complex of relations both among themselves andwith those above and below. The lower the stratum,the deeper (and in general the more di�cult) the idea.Thus the idea of an ‘irrational’ is deeper than that of aninteger; and Pythagoras’s theorem is, for that reason,deeper than Euclid’s.

Let us concentrate our attention on the relations be-tween the integers, or some other group of objects lyingin some particular stratum. Then it may happen thatone of these relations can be comprehended completely,that we can recognize and prove, for example, someproperty of the integers, without any knowledge of thecontents of lower strata. Thus we proved Euclid’s theo-rem by consideration of properties of integers only. Butthere are also many theorems about integers which we

81

cannot appreciate properly, and still less prove, withoutdigging deeper and considering what happens below.

It is easy to �nd examples in the theory of primenumbers. Euclid’s theorem is very important, but notvery deep: we can prove that there are in�nitely manyprimes without using any notion deeper than that of‘divisibility’. But new questions suggest themselves assoon as we know the answer to this one. There is anin�nity of primes, but how is this in�nity distributed ?Given a large number N , say 1080 or 101010 , about howmany primes are there less than N ?13 When we askthese questions, we �nd ourselves in a quite di�erentposition. We can answer them, with rather surpris-ing accuracy, but only by boring much deeper, leavingthe integers above us for a while, and using the mostpowerful weapons of the modern theory of functions.Thus the theorem which answers our questions (theso-called ‘Prime Number Theorem’) is a much deepertheorem than Euclid’s or even Pythagoras’s.

I could multiply examples, but this notion of ‘depth’is an elusive one even for a mathematician who canrecognize it, and I can hardly suppose that I could sayanything more about it here which would be of muchhelp to other readers.

13It is supposed that the number of protons in the universe isabout 1080. The number 1010

10, if written at length, would occupy

about 50,000 volumes of average size.As I mentioned in §14, there are 50,847,478 primes less than

1,000,000,000; but that is as far as our exact knowledge extends.

82

18

There is still one point remaining over from §11, whereI started the comparison between ‘real mathematics’and chess. We may take it for granted now that insubstance, seriousness, signi�cance, the advantage ofthe real mathematical theorem is overwhelming. It isalmost equally obvious, to a trained intelligence, thatit has a great advantage in beauty also; but this advan-tage is much harder to de�ne or locate, since the maindefect of the chess problem is plainly its ‘triviality’, andthe contrast in this respect mingles with and disturbsany more purely aesthetic judgment. What ‘purely aes-thetic’ qualities can we distinguish in such theoremsas Euclid’s and Pythagoras’s? I will not risk more thana few disjointed remarks.

In both theorems (and in the theorems, of course, Iinclude the proofs) there is a very high degree of un-expectedness, combined with inevitability and economy.The arguments take so odd and surprising a form; theweapons used seem so childishly simple when com-pared with the far-reaching results; but there is noescape from the conclusions. There are no complica-tions of detail—one line of attack is enough in eachcase; and this is true too of the proofs of many muchmore di�cult theorems, the full appreciation of whichdemands quite a high degree of technical pro�ciency.We do not want many ‘variations’ in the proof of amathematical theorem: ‘enumeration of cases’, indeed,is one of the duller forms of mathematical argument.

83

A mathematical proof should resemble a simple andclear-cut constellation, not a scattered cluster in theMilky Way.

A chess problem also has unexpectedness, and acertain economy; it is essential that the moves shouldbe surprising, and that every piece on the board shouldplay its part. But the aesthetic e�ect is cumulative. Itis essential also (unless the problem is too simple to bereally amusing) that the key-move should be followedby a good many variations, each requiring its ownindividual answer. ‘If P-B5 then Kt-R6; if . . . . then. . . . ; if . . . . then . . . . ’—the e�ect would be spoilt ifthere were not a good many di�erent replies. All thisis quite genuine mathematics, and has its merits; butit is just that ‘proof by enumeration of cases’ (and ofcases which do not, at bottom, di�er at all profoundly14)which a real mathematician tends to despise.

I am inclined to think that I could reinforce my ar-gument by appealing to the feelings of chess-playersthemselves. Surely a chess master, a player of greatgames and great matches, at bottom scorns a prob-lemist’s purely mathematical art. He has much of it inreserve himself, and can produce it in an emergency: ‘ifhe had made such and such a move, then I had such andsuch a winning combination in mind.’ But the ‘greatgame’ of chess is primarily psychological, a con�ictbetween one trained intelligence and another, and nota mere collection of small mathematical theorems.

14I believe that it is now regarded as a merit in a problem thatthere should be many variations of the same type.

84

19

I must return to my Oxford apology, and examine a lit-tle more carefully some of the points which I postponedin §6. It will be obvious by now that I am interestedin mathematics only as a creative art. But there areother questions to be considered, and in particular thatof the ‘utility’ (or uselessness) of mathematics, aboutwhich there is much confusion of thought. We mustalso consider whether mathematics is really quite so‘harmless’ as I took for granted in my Oxford lecture.

A science or an art may be said to be ‘useful’ if itsdevelopment increases, even indirectly, the materialwell-being and comfort of men, if it promotes happi-ness, using that word in a crude and commonplace way.Thus medicine and physiology are useful because theyrelieve su�ering, and engineering is useful because ithelps us to build houses and bridges, and so to raise thestandard of life (engineering, of course, does harm aswell, but that is not the question at the moment). Nowsome mathematics is certainly useful in this way; theengineers could not do their job without a fair work-ing knowledge of mathematics, and mathematics isbeginning to �nd applications even in physiology. Sohere we have a possible ground for a defense of math-ematics; it may not be the best, or even a particularlystrong defense, but it is one which we must examine.The ‘nobler’ uses of mathematics, if such they be, theuses which it shares with all creative art, will be irrele-vant to our examination. Mathematics may, like poetry

85

or music, ‘promote and sustain a lofty habit of mind’,and so increase the happiness of mathematicians andeven of other people; but to defend it on that groundwould be merely to elaborate what I have said already.What we have to consider now is the ‘crude’ utility ofmathematics.

20

All this may seem very obvious, but even here there isoften a good deal of confusion, since the most ‘useful’subjects are quite commonly just those which it is mostuseless for most of us to learn. It is useful to have anadequate supply of physiologists and engineers; butphysiology and engineering are not useful studies forordinary men (though their study may of course bedefended on other grounds). For my own part I havenever once found myself in a position where such scien-ti�c knowledge as I possess, outside pure mathematics,has brought me the slightest advantage.

It is indeed rather astonishing how little practi-cal value scienti�c knowledge has for ordinary men,how dull and commonplace such of it as has value is,and how its value seems almost to vary inversely toits reputed utility. It is useful to be tolerably quick atcommon arithmetic (and that, of course, is pure mathe-matics). It is useful to know a little French or German,a little history and geography, perhaps even a little eco-nomics. But a little chemistry, physics, or physiologyhas no value at all in ordinary life. We know that the

86

gas will burn without knowing its constitution; whenour cars break down we take them to a garage; whenour stomach is out of order, we go to a doctor or adrugstore. We live either by rule of thumb or on otherpeople’s professional knowledge.

However, this is a side issue, a matter of pedagogy,interesting only to schoolmasters who have to adviseparents clamoring for a ‘useful’ education for theirsons. Of course we do not mean, when we say thatphysiology is useful, that most people ought to studyphysiology, but that the development of physiologyby a handful of experts will increase the comfort ofthe majority. The questions which are important forus now are, how far mathematics can claim this sortof utility, what kinds of mathematics can make thestrongest claims, and how far the intensive study ofmathematics, as it is understood by mathematicians,can be justi�ed on this ground alone.

21

It will probably be plain by now to what conclusions Iam coming; so I will state them at once dogmaticallyand then elaborate them a little. It is undeniable thata good deal of elementary mathematics—and I use theword ‘elementary’ in the sense in which professionalmathematicians use it, in which it includes, for example,a fair working knowledge of the di�erential and inte-gral calculus—has considerable practical utility. Theseparts of mathematics are, on the whole, rather dull;

87

they are just the parts which have least aesthetic value.The ‘real’ mathematics of the ‘real’ mathematicians,the mathematics of Fermat and Euler and Gauss andAbel and Riemann, is almost wholly ‘useless’ (and thisis as true of ‘applied’ as of ‘pure’ mathematics) . It isnot possible to justify the life of any genuine profes-sional mathematician on the ground of the ‘utility’ ofhis work.

But here I must deal with a misconception. It issometimes suggested that pure mathematicians gloryin the uselessness of their work, and make it a boastthat it has no practical applications.15 The imputationis usually based on an incautious saying attributed toGauss, to the e�ect that, if mathematics is the queen ofthe sciences, then the theory of numbers is, because ofits supreme uselessness, the queen of mathematics—Ihave never been able to �nd an exact quotation. I amsure that Gauss’s saying (if indeed it be his) has beenrather crudely misinterpreted. If the theory of num-bers could be employed for any practical and obviouslyhonorable purpose, if it could be turned directly to thefurtherance of human happiness or the relief of hu-man su�ering, as physiology and even chemistry can,then surely neither Gauss nor any other mathematician

15I have been accused of taking this view myself. I once said that‘a science is said to be useful if its development tends to accentuatethe existing inequalities in the distribution of wealth, or moredirectly 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 �ourish, though one perhapsexcusable at the time when it was written.

88

would have been so foolish as to decry or regret suchapplications. But science works for evil as well as forgood (and particularly, of course, in time of war); andboth Gauss and lesser mathematicians may be justi�edin rejoicing that there is one science at any rate, andthat their own, whose very remoteness from ordinaryhuman activities should keep it gentle and clean.

22

There is another misconception against which we mustguard. It is quite natural to suppose that there is a greatdi�erence in utility between ‘pure’ and ‘applied’ math-ematics. This is a delusion: there is a sharp distinctionbetween the two kinds of mathematics, which I willexplain in a moment, but it hardly a�ects their utility.

How do pure and applied mathematics di�er fromone another? This is a question which can be answeredde�nitely and about which there is general agreementamong mathematicians. There will be nothing in theleast unorthodox about my answer, but it needs a littlepreface.

My next two sections will have a mildly philosoph-ical �avor. The philosophy will not cut deep, or bein any way vital to my main theses; but I shall usewords which are used very frequently with de�nitephilosophical implications, and a reader might wellbecome confused if I did not explain how I shall usethem.

89

I have often used the adjective ‘real’, and as weuse it commonly in conversation. I have spoken of‘real mathematics’ and ‘real mathematicians’, as I mighthave spoken of ‘real poetry’ or ‘real poets’, and I shallcontinue to do so. But I shall also use the word ‘reality’,and with two di�erent connotations.

In the �rst place, I shall speak of ‘physical reality’,and here again I shall be using the word in the ordinarysense. By physical reality I mean the material world,the world of day and night, earthquakes and eclipses,the world which physical science tries to describe.

I hardly suppose that, up to this point, any readeris likely to �nd trouble with my language, but now Iam near to more di�cult ground. For me, and I sup-pose for most mathematicians, there is another reality,which I will call ‘mathematical reality’; and there is nosort of agreement about the nature of mathematicalreality among either mathematicians or philosophers.Some hold that it is ‘mental’ and that in some sense weconstruct it, others that it is outside and independentof us. A man who could give a convincing account ofmathematical reality would have solved very many ofthe most di�cult problems of metaphysics. If he couldinclude physical reality in his account, he would havesolved them all.

I should not wish to argue any of these questionshere even if I were competent to do so, but I will statemy own position dogmatically in order to avoid minormisapprehensions. I believe that mathematical realitylies outside us, that our function is to discover or observeit, and that the theorems which we prove, and which we

90

describe grandiloquently as our ‘creations’, are simplyour notes of our observations. This view has beenheld, in one form or another, by many philosophersof high reputation from Plato onwards, and I shall usethe language which is natural to a man who holds it.A reader who does not like the philosophy can alterthe language: it will make very little di�erence to myconclusions.

23

The contrast between pure and applied mathematicsstands out most clearly, perhaps, in geometry. There isthe science of pure geometry*, in which there are manygeometries, projective geometry, Euclidean geometry,non-Euclidean geometry, and so forth.16 Each of thesegeometries is a model, a pattern of ideas, and is to bejudged by the interest and beauty of its particular pat-tern. It is a map or picture, the joint product of manyhands, a partial and imperfect copy (yet exact so far asit extends) of a section of mathematical reality. But thepoint which is important to us now is this, that there isone thing at any rate of which pure geometries are notpictures, and that is the spatio-temporal reality of thephysical world. It is obvious, surely, that they cannotbe, since earthquakes and eclipses are not mathematicalconcepts.

16We must of course, for the purposes of this discussion, countas pure geometry what mathematicians call ‘analytical’ geometry.

91

This may sound a little paradoxical to an outsider,but it is a truism to a geometer; and I may perhaps beable to make it clearer by an illustration. Let us supposethat I am giving a lecture on some system of geometry,such as ordinary Euclidean geometry, and that I draw�gures on the blackboard to stimulate the imaginationof my audience, rough drawings of straight lines orcircles or ellipses. It is plain, �rst, that the truth of thetheorems which I prove is in no way a�ected by thequality of my drawings. Their function is merely tobring home my meaning to my hearers, and, if I can dothat, there would be no gain in having them redrawn bythe most skillful draughtsman. They are pedagogicalillustrations, not part of the real subject-matter of thelecture.

Now let us go a stage further. The room in whichI am lecturing is part of the physical world, and hasitself a certain pattern. The study of that pattern, andof the general pattern of physical reality, is a science initself, which we may call * physical geometry’. Supposenow that a violent dynamo, or a massive gravitatingbody, is introduced into the room. Then the physiciststell us that the geometry of the room is changed, itswhole physical pattern slightly but de�nitely distorted.Do the theorems which I have proved become false?Surely it would be nonsense to suppose that the proofsof them which I have given are a�ected in any way. Itwould be like sup-posing that a play of Shakespeare ischanged when a reader spills his tea over a page. Theplay is independent of the pages on which it is printed,

92

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, nat-urally take a di�erent view, since they are preoccupiedwith the physical world itself, which also has its struc-ture or pattern. We cannot describe this pattern exactly,as we can that of a pure geometry, but we can say some-thing signi�cant about it. We can describe, sometimesfairly accurately, sometimes very roughly, the relationswhich hold between some of its constituents, and com-pare them with the exact relations holding betweenconstituents of some system of pure geometry. Wemay be able to trace a certain resemblance betweenthe two sets of relations, and then the pure geometrywill become interesting to physicists; it will give us, tothat extent, a map which ‘�ts the facts’ of the physicalworld. The geometer o�ers to the physicist a wholeset of maps from which to choose. One map, perhaps,will �t the facts better than others, and then the ge-ometry which provides that particular map will be thegeometry most important for applied mathematics. Imay add that even a pure mathematician may �nd hisappreciation of this geometry quickened, since there isno mathematician so pure that he feels no interest atall in the physical world; but, in so far as he succumbsto this temptation, he will be abandoning his purelymathematical position.

93

24

There is another remark which suggests itself hereand which physicists may �nd paradoxical, though theparadox will probably seem a good deal less than it dideighteen years ago. I will express it in much the samewords which I used in 1922 in an address to SectionA of the British Association. My audience then wascomposed almost entirely of physicists, and I may havespoken a little provocatively on that account; but Iwould still stand by the substance of what I said.

I began by saying that there is probably less di�er-ence between the positions of a mathematician and of aphysicist than is generally supposed, and that the mostimportant seems to me to be this, that the mathemati-cian is in much more direct contact with reality. Thismay seem a paradox, since it is the physicist who dealswith the subject-matter usually described as ‘real’; buta very little re�ection is enough to show that the physi-cist’s reality, whatever it may be, has few or none ofthe attributes which common sense ascribes instinc-tively to reality. A chair may be a collection of whirlingelectrons, or an idea in the mind of God: each of theseaccounts of it may have its merits, but neither conformsat all closely to the suggestions of common sense.

I went on to say that neither physicists nor philoso-phers have ever given any convincing account of what‘physical reality’ is, or of how the physicist passes, fromthe confused mass of fact or sensation with which hestarts, to the construction of the objects which he calls

94

‘real’. Thus we cannot be said to know what the subject-matter of physics is; but this need not prevent us fromunderstanding roughly what a physicist is trying to do.It is plain that he is trying to correlate the incoherentbody of crude fact confronting him with some de�niteand orderly scheme of abstract relations, the kind ofscheme which he can borrow only from mathematics.

A mathematician, on the other hand, is workingwith his own mathematical reality. Of this reality, as Iexplained in §22,1 take a ‘realistic’ and not an ‘idealistic’view. At any rate (and this was my main point) thisrealistic view is much more plausible of mathematicalthan of physical reality, because mathematical objectsare so much more what they seem. A chair or a star isnot in the least like what it seems to be; the more wethink of it, the fuzzier its outlines become in the hazeof sensation which surrounds it; but ‘2’ or ‘317’ hasnothing to do with sensation, and its properties standout the more clearly the more closely we scrutinizeit. It may be that modern physics �ts best into someframework of idealistic philosophy—I do not believeit, but there are eminent physicists who say so. Puremathematics, on the other hand, seems to me a rock onwhich all idealism founders: 317 is a prime, not becausewe think so, or because our minds are shaped in oneway rather than another, but because it is so, becausemathematical reality is built that way.

95

25

These distinctions between pure and applied mathe-matics are important in themselves, but they have verylittle bearing on our discussion of the ‘usefulness’ ofmathematics. I spoke in §21 of the ‘real’ mathematicsof Fermat and other great mathematicians, the math-ematics which has permanent aesthetic value, as forexample the best Greek mathematics has, the mathe-matics which is eternal because the best of it may, likethe best literature, continue to cause intense emotionalsatisfaction to thousands of people after thousands ofyears. These men were all primarily pure mathemati-cians (though the distinction was naturally a good dealless sharp in their days than it is now); but I was notthinking only of pure mathematics. I count Maxwelland Einstein, Eddington and Dirac, among ‘real’ math-ematicians. The great modern achievements of appliedmathematics have been in relativity and quantum me-chanics, and these subjects are, at present at any rate,almost as ‘useless’ as the theory of numbers. It is thedull and elementary parts of applied mathematics, asit is the dull and elementary parts of pure mathemat-ics, that work for good or ill. Time may change allthis. No one foresaw the applications of matrices andgroups and other purely mathematical theories to mod-ern physics, and it may be that some of the ‘highbrow’applied mathematics will become ‘useful’ in as unex-pected a way; but the evidence so far points to theconclusion that, in one subject as in the other, it is

96

what is commonplace and dull that counts for practicallife.

I can remember Eddington giving a happy exampleof the unattractiveness of ‘useful’ science. The BritishAssociation held a meeting in Leeds, and it was thoughtthat the members might like to hear something of theapplications of science to the ‘heavy woolen’ indus-try. But the lectures and demonstrations arranged forthis purpose were rather a �asco. It appeared that themembers (whether citizens of Leeds or not) wanted tobe entertained, and that ‘heavy wool’ is not at all anentertaining subject. So the attendance at these lec-tures was very disappointing; but those who lecturedon the excavations at Knossos, or on relativity, or onthe theory of prime numbers, were delighted by theaudiences that they drew.

26

What parts of mathematics are useful?First, the bulk of school mathematics, arithmetic,

elementary algebra, elementary Euclidean geometry,elementary di�erential and integral calculus. We mustexcept a certain amount of what is taught to ‘specialists’,such as projective geometry. In applied mathematics,the elements of mechanics (electricity, as taught inschools, must be classi�ed as physics).

Next, a fair proportion of university mathematics isalso useful, that part of it which is really a developmentof school mathematics with a more �nished technique,

97

and a certain amount of the more physical subjectssuch as electricity and hydromechanics. We must alsoremember that a reserve of knowledge is always an ad-vantage, and that the most practical of mathematiciansmay be seriously handicapped if his knowledge is thebare minimum which is essential to him; and for thisreason we must add a little under every heading. Butour general conclusion must be that such mathematicsis useful as is wanted by a superior engineer or a mod-erate physicist; and that is roughly the same thing asto say, such mathematics as has no particular aestheticmerit. Euclidean geometry, for example, is useful in sofar as it is dull—we do not want the axiomatics of par-allels, or the theory of proportion, or the constructionof the regular pentagon.

One rather curious conclusion emerges, that puremathematics is on the whole distinctly more usefulthan applied. A pure mathematician seems to have theadvantage on the practical as well as on the aestheticside. For what is useful above all is technique, andmathematical technique is taught mainly through puremathematics.

I hope that I need not say that I am not trying todecry mathematical physics, a splendid subject withtremendous problems where the �nest imaginationshave run riot. But is not the position of an ordinaryapplied mathematician in some ways a little pathetic? Ifhe wants to be useful, he must work in a humdrum way,and he cannot give full play to his fancy even when hewishes to rise to the heights. ‘Imaginary’ universes areso much more beautiful than this stupidly constructed

98

‘real’ one; and most of the �nest products of an appliedmathematician’s fancy must be rejected, as soon as theyhave been created, for the brutal but su�cient reasonthat they do not �t the facts.

The general conclusion, surely, stands out plainlyenough. If useful knowledge is, as we agreed provi-sionally to say, knowledge which is likely, now or inthe comparatively near future, to contribute to the ma-terial comfort of mankind, so that mere intellectualsatisfaction is irrelevant, then the great bulk of highermathematics is useless. Modern geometry and algebra,the theory of numbers, the theory of aggregates andfunctions, relativity, quantum mechanics—no one ofthem stands the test much better than another, andthere is no real mathematician whose life can be jus-ti�ed on this ground. If this be the test, then Abel,Riemann, and Poincare wasted their lives; their contri-bution to human comfort was negligible, and the worldwould have been as happy a place without them.

27

It may be objected that my concept of ‘utility’ has beentoo narrow, that I have de�ned it in terms of ‘happi-ness’ or ‘comfort’ only, and have ignored the general‘social’ e�ects of mathematics on which recent writ-ers, with very di�erent sympathies, have laid so muchstress. Thus Whitehead (who has been a mathemati-cian) speaks of ‘the tremendous e�ect of mathematicalknowledge on the lives of men, on their daily avoca-

99

tions, on the organization of society’; and Hogben (whois as unsympathetic to what I and other mathemati-cians call mathematics as Whitehead is sympathetic)says that ‘without a knowledge of mathematics, thegrammar of size and order, we cannot plan the rationalsociety in which there will be leisure for all and povertyfor none’ (and much more to the same e�ect).

I cannot really believe that all this eloquence willdo much to comfort mathematicians. The language ofboth writers is violently exaggerated, and both of themignore very obvious distinctions. This is very naturalin Hogben’s case, since he is admittedly not a mathe-matician; he means by ‘mathematics’ the mathemat-ics which he can understand, and which I have called‘school’ mathematics. This mathematics has many uses,which I have admitted, which we can call ‘social’ if weplease, and which Hogben enforces with many inter-esting appeals to the history of mathematical discovery.It is this which gives his book its merit, since it enableshim to make plain, to many readers who never havebeen and never will be mathematicians, that there ismore in mathematics than they thought. But he hashardly any understanding of ‘real’ mathematics (as anyone who reads what he says about Pythagoras’s the-orem, or about Euclid and Einstein, can tell at once),and still less sympathy with it (as he spares no pains toshow). ‘Real’ mathematics is to him merely an objectof contemptuous pity.

It is not lack of understanding or of sympathy whichis the trouble in Whitehead’s case; but he forgets, inhis enthusiasm, distinctions with which he is quite fa-

100

miliar. The mathematics which has this ‘tremendouse�ect’ on the ‘daily avocations of men’ and on ‘theorganization of society’ is not the Whitehead but theHogben mathematics. The mathematics which canbe used ‘for ordinary purposes by ordinary men’ isnegligible, and that which can be used by economistsor sociologists hardly rises to ‘scholarship standard’.The Whitehead mathematics may a�ect astronomy orphysics profoundly, philosophy very appreciably—highthinking of one kind is always likely to a�ect highthinking of another—but it has extremely little e�ecton anything else. Its ‘tremendous e�ects’ have been,not on men generally, but on men like Whitehead him-self.

28

There are then two mathematics. There is the realmathematics of the real mathematicians, and there iswhat I will call the ‘trivial’ mathematics, for want of abetter word. The trivial mathematics may be justi�edby arguments which would appeal to Hogben, or otherwriters of his school, but there is no such defense forthe real mathematics, which must be justi�ed as art ifit can be justi�ed at all. There is nothing in the leastparadoxical or unusual in this view, which is that heldcommonly by mathematicians.

We have still one more question to consider. Wehave concluded that the trivial mathematics is, on thewhole, useful, and that the real mathematics, on the

101

whole, is not; that the trivial mathematics does, and thereal mathematics does not, ‘do good’ in a certain sense;but we have still to ask whether either sort of mathe-matics does harm. It would be paradoxical to suggestthat mathematics of any sort does much harm in timeof peace, so that we are driven to the consideration ofthe e�ects of mathematics on war. It is very di�cultto argue such questions at all dispassionately now, andI should have preferred to avoid them; but some sortof discussion seems inevitable. Fortunately, it need notbe a long one.

There is one comforting conclusion which is easyfor a real mathematician. Real mathematics has no ef-fects on war. No one has yet discovered any warlikepurpose to be served by the theory of numbers or rela-tivity, and it seems very unlikely that anyone will doso for many years. It is true that there are branches ofapplied mathematics, such as ballistics and aerodynam-ics, which have been developed deliberately for warand demand a quite elaborate technique: it is perhapshard to call them ‘trivial’, but none of them has anyclaim to rank as ‘real’. They are indeed repulsively uglyand intolerably dull; even Littlewood could not makeballistics respectable, and if he could not who can? Soa real mathematician has his conscience clear; thereis nothing to be set against any value his work mayhave; mathematics is, as I said at Oxford, a ‘harmlessand innocent’ occupation.

The trivial mathematics, on the other hand, hasmany applications in war. The gunnery experts andaeroplane designers, for example, could not do their

102

work without it. And the general e�ect of these ap-plications is plain: mathematics facilitates (if not soobviously as physics or chemistry) modem, scienti�c,‘total’ war.

It is not so clear as it might seem that this is to beregretted, since there are two sharply contrasted viewsabout modern scienti�c war. The �rst and the mostobvious is that the e�ect of science on war is merelyto magnify its horror, both by increasing the su�eringsof the minority who have to �ght and by extendingthem to other classes. This is the most natural andthe orthodox view. But there is a very di�erent viewwhich seems also quite tenable, and which has beenstated with great force by Haldane in Callinicus.17 Itcan be maintained that modern warfare is less horriblethan the warfare of pre-scienti�c times; that bombs areprobably more merciful than bayonets; that lachryma-tory gas and mustard gas are perhaps the most humaneweapons yet devised by military science; and that theorthodox view rests solely on loose-thinking sentimen-talism.18 It may also be urged (though this was not oneof Haldane’s theses) that the equalization of risks whichscience was expected to bring would be in the long runsalutary; that a civilian’s life is not worth more than asoldier’s, nor a woman’s than a man’s; that anything is

17J.B.S. Haldane, Callinicus: a defense of Chemical Warfare (1924)18I do not wish to prejudge the question by this much misused

word; it may be used quite legitimately to indicate certain types ofunbalanced emotion. Many people, of course, use ‘sentimentalism’as a term of abuse for other people’s decent feelings, and ‘realism’as a disguise for their own brutality.

103

better than the concentration of savagery on one par-ticular class; and that, in short, the sooner war comes‘all out’ the better.

I do not know which of these views is nearer tothe truth. It is an urgent and a moving question, butI need not argue it here. It concerns only the ‘trivial’mathematics, which it would be Hogben’s business todefend rather than mine. The case for his mathematicsmay be rather more than a little soiled; the case formine is una�ected.

Indeed, there is more to be said, since there is onepurpose at any rate which the real mathematics mayserve in war. When the world is mad, a mathematicianmay �nd in mathematics an incomparable anodyne. Formathematics is, of all the arts and sciences, the most aus-tere and the most remote, and a mathematician shouldbe of all men the one who can most easily take refugewhere, as Bertrand Russell says, ‘one at least of ournobler impulses can best escape from the dreary exileof the actual world’. It is a pity that it should be nec-essary to make one very serious reservation—he mustnot be too old. Mathematics is not a contemplative buta creative subject; no one can draw much consolationfrom it when he has lost the power or the desire tocreate; and that is apt to happen to a mathematicianrather soon. It is a pity, but in that case he does notmatter a great deal anyhow, and it would be silly tobother about him.

104

29

I will end with a summary of my conclusions, butputting them in a more personal way. I said at thebeginning that anyone who defends his subject will�nd that he is defending himself; and my justi�cationof the life of a professional mathematician is boundto be, at bottom, a justi�cation of my own. Thus thisconcluding section will be in its substance a fragmentof autobiography.

I cannot remember ever having wanted to be any-thing but a mathematician. I suppose that it was alwaysclear that my speci�c abilities lay that way, and it neveroccurred to me to question the verdict of my elders. Ido not remember having felt, as a boy, any passion formathematics, and such notions as I may have had ofthe career of a mathematician were far from noble. Ithought of mathematics in terms of examinations andscholarships: I wanted to beat other boys, and thisseemed to be the way in which I could do so mostdecisively.

I was about �fteen when (in a rather odd way) myambitions took a sharper turn. There is a book by ‘AlanSt Aubyn’ called A Fellow of Trinity, one of a seriesdealing with what is supposed to be Cambridge collegelife.19 I suppose that it is a worse book than most ofMarie Corelli’s; but a book can hardly be entirely badif it �res a clever boy’s imagination. There are two

19‘Alan St Aubyn’ was Mrs Frances Marshall, wife of MatthewMarshall.

105

heroes, a primary hero called Flowers, who is almostwholly good, and a secondary hero, a much weakervessel, called Brown. Flowers and Brown �nd manydangers in university life, but the worst is a gamblingsaloon in Chesterton run by the Misses Bellenden, twofascinating but extremely wicked young ladies.20 Flow-ers survives all these troubles, is Second Wrangler andSenior Classic, and succeeds automatically to a Fellow-ship (as I suppose he would have done then). Brownsuccumbs, ruins his parents, takes to drink, is savedfrom delirium tremens during a thunderstorm only bythe prayers of the Junior Dean, has much di�culty inobtaining even an Ordinary Degree, and ultimately be-comes a missionary. The friendship is not shatteredby these unhappy events, and Flowers’s thoughts strayto Brown, with a�ectionate pity, as he drinks port andeats walnuts for the �rst time in Senior CombinationRoom.

Now Flowers was a decent enough fellow (so faras ‘Alan St Aubyn’ could draw one), but even my unso-phisticated mind refused to accept him as clever. If hecould do these things, why not I? In particular, the �nalscene in Combination Room fascinated me completely,and from that time, until I obtained one, mathematicsmeant to me primarily a Fellowship of Trinity.

I found at once, when I came to Cambridge, thata Fellowship implied original work’, but it was a longtime before I formed any de�nite idea of research. I hadof course found at school, as every future mathemati-

20Actually, Chesterton lacks picturesque features.

106

cian does, that I could often do things much better thanmy teachers; and even at Cambridge I found, thoughnaturally much less frequently, that I could sometimesdo things better than the College lecturers. But I was re-ally quite ignorant, even when I took the Tripos, of thesubjects on which I have spent the rest of my life; andI still thought of mathematics as essentially a ‘competi-tive’ subject. My eyes were �rst opened by ProfessorLove, who taught me for a few terms and gave memy �rst serious conception of analysis. But the greatdebt which I owe to him—he was, after all, primarilyan applied mathematician—was his advice to read Jor-dan’s famous Cours d’analyse; and I shall never forgetthe astonishment with which I read that remarkablework, the �rst inspiration for so many mathematiciansof my generation, and learned for the �rst time as Iread it what mathematics really meant. From that timeonwards I was in my way a real mathematician, withsound mathematical ambitions and a genuine passionfor mathematics.

I wrote a great deal during the next ten years, butvery little of any importance; there are not more thanfour or �ve papers which I can still remember withsome satisfaction. The real crises of my career cameten or twelve years later, in 1911, when I began mylong collaboration with Littlewood, and in 1913, whenI discovered Ramanujan. All my best work since thenhas been bound up with theirs, and it is obvious thatmy association with them was the decisive event of mylife. I still say to myself when I am depressed, and �ndmyself forced to listen to pompous and tiresome people,

107

‘Well, I have done one thing you could never have done,and that is to have collaborated with both Littlewoodand Ramanujan on something like equal terms.’ It isto them that I owe an unusually late maturity: I wasat my best at a little past forty, when I was a professorat Oxford. Since then I have su�ered from that steadydeterioration which is the common fate of elderly menand particularly of elderly mathematicians. A mathe-matician may still be competent enough at sixty, but itis useless to expect him to have original ideas.

It is plain now that my life, for what it is worth,is �nished, and that nothing I can do can perceptiblyincrease or diminish its value. It is very di�cult to bedispassionate, but I count it a ‘success’; I have had morereward and not less than was due to a man of my partic-ular grade of ability. I have held a series of comfortableand ‘digni�ed’ positions. I have had very little troublewith the duller routine of universities. I hate ‘teach-ing’, and have had to do very* little, such teaching as Ihave done having been almost entirely supervision ofresearch; I love lecturing, and have lectured a great dealto extremely able classes; and I have always had plentyof leisure for the researches which have been the onegreat permanent happiness of my life. I have found iteasy to work with others, and have collaborated on alarge scale with two exceptional mathematicians; andthis has enabled me to add to mathematics a good dealmore than I could reasonably have expected. I havehad my disappointments, like any other mathematician,but none of them has been too serious or has made meparticularly unhappy. If I had been o�ered a life nei-

108

ther better nor worse when I was twenty, I would haveaccepted without hesitation.

It seems absurd to suppose that I could have ‘donebetter’. I have no linguistic or artistic ability, and verylittle interest in experimental science. I might havebeen a tolerable philosopher, but not one of a very orig-inal kind. I think that I might have made a good lawyer;but journalism is the only profession, outside academiclife, in which I should have felt really con�dent of mychances. There is no doubt that I was right to be a math-ematician, if the criterion is to be what is commonlycalled success.

My choice was right, then, if what I wanted was areasonably comfortable and happy life. But solicitorsand stockbrokers and bookmakers often lead comfort-able and happy lives, and it is very di�cult to see howthe world is the richer for their existence. Is there anysense in which I can claim that my life has been lessfutile than theirs? It seems to me again that there isonly one possible answer: yes, perhaps, but, if so, forone reason only.

I have never done anything ‘useful’. No discoveryof mine has made, or is likely to make, directly or indi-rectly, for good or ill, the least di�erence to the amenityof the world. I have helped to train other mathemati-cians, but mathematicians of the same kind as myself,and their work has been, so far at any rate as I havehelped them to it, as useless as my own. Judged by allpractical standards, the value of my mathematical life isnil; and outside mathematics it is trivial anyhow. I havejust one chance of escaping a verdict of complete trivi-

109

ality, that I may be judged to have created somethingworth creating. And that I have created something isundeniable: the question is about its value.

The case for my life, then, or for that of any oneelse who has been a mathematician in the same sensein which I have been one, is this: that I have addedsomething to knowledge, and helped others to addmore; and that these somethings have a value whichdi�ers in degree only, and not in kind, from that of thecreations of the great mathematicians, or of any of theother artists, great or small, who have left some kindof memorial behind them.

110

Note

Professor Broad and Dr Snow have both remarked tome that, if I am to strike a fair balance between the goodand evil done by science, I must not allow myself to betoo much obsessed by its e�ects on war; and that, evenwhen I am thinking of them, I must remember that ithas many very important e�ects besides those whichare purely destructive. Thus (to take the latter point�rst), I must re-member (a) that the organization of anentire population for war is only possible through sci-enti�c methods; (b) that science has greatly increasedthe power of propaganda, which is used almost exclu-sively for evil; and (c) that it has made ‘neutrality5almost impossible or unmeaning, so that there are nolonger ‘islands of peace5 from which sanity and restora-tion might spread out gradually after war. All this, ofcourse, tends to reinforce the case against science. Onthe other hand, even if we press this case to the utmost,it is hardly possible to maintain seriously that the evildone by science is not altogether outweighed by thegood. For example, if ten million lives were lost in ev-ery war, the net e�ect of science would still have been

to increase the average length of life. In short, my §28is much too ‘sentimental’.

I do not dispute the justice of these criticisms, but,for the reasons which I state in my preface, I have foundit impossible to meet them in my text, and contentmyself with this acknowledgment.

Dr Snow has also made an interesting minor pointabout §8. Even if we grant that ‘Archimedes will beremembered when Aeschylus is forgotten’, is not math-ematical fame a little too ‘anonymous’ to be whollysatisfying? We could form a fairly coherent picture ofthe personality of Aeschylus (still more, of course, ofShakespeare or Tolstoi) from their works alone, whileArchimedes and Eudoxus would remain mere names.

Mr J.M. Lomas put this point more picturesquelywhen we were passing the Nelson column in TrafalgarSquare. If I had a statue on a column in London, wouldI prefer the column to be so high that the statue wasinvisible, or low enough for the features to be recog-nizable? I would choose the �rst alternative, Dr Snow,presumably, the second.

112