A Mathematician's Apology PDF
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A Mathematician's Apology Godfrey Harold Hardy A Mathematician's Apology Godfrey Harold Hardy BAALTIS PUBLISHING foreword ItwasaperfectlyordinarynightatChrist’shightable, exceptthatHardywasdiningasaguest. Hehadjust returned to Cambridge as Sadleirian professor, and I had heard something of him from young Cambridge mathematicians.Theyweredelightedtohavehimback: hewasareal mathematician,theysaid,notlikethose Diracs and Bohrs the physicists were always talking about: he was the purest of the pure. He was also unorthodox,eccentric,radical,readytotalkaboutany- thing. This was 1931, and the phrase was not yet in Englishuse,butinlaterdaystheywouldhavesaidthat insomeindefinablewayhehadstarquality. So,fromlowerdownthetable,Ikeptstudyinghim. He was then in his early fifties: his hair was already gray, above skin so deeply sunburnt that it stayed a kindofRedIndianbronze. Hisfacewasbeautiful—high cheekbones,thinnose,spiritualandausterebutcapa- bleofdissolvingintoconvulsionsofinternalgamin-like amusement. He had opaque brown eyes, bright as a bird’s—akindofeyenotuncommonamongthosewith agiftforconceptualthought. Cambridgeatthattime wasfullofunusualanddistinguishedfaces—buteven then,Ithoughtthatnight,Hardy’sstoodout. I do not remember what he was wearing. It may easilyhavebeenasportscoatandgreyflannelsunder hisgown. LikeEinstein,hedressedtopleasehimself: though,unlikeEinstein,hediversifiedhiscasualcloth- ingbyatasteforexpensivesilkshirts. Aswesatroundthecombination-roomtable,drink- ingwineafterdinner,someonesaidthatHardywanted totalktomeaboutcricket. Ihadbeenelectedonlya yearbefore,butChrist’swasthenasmallcollege,and thepastimesofeventhejuniorfellowsweresooniden- tified. Iwastakentositbyhim. Iwasnotintroduced. He was, as I later discovered, shy and self-conscious inallformalactions,andhadadreadofintroductions. He just put his head down as it were in a butt of ac- knowledgment,andwithoutanypreamblewhat-ever began: ‘You’resupposedtoknowsomethingaboutcricket, aren’tyou?’ Yes,Isaid,Iknewabit. Immediatelyhebegantoputmethroughamoder- atelystiffviva. DidIplay? Whatsortofperformerwas I?Ihalf-guessedthathehadahorrorofpersons,then prevalentinacademicsociety,whodevotedlystudied theliteraturebuthadneverplayedthegame. Itrotted out my credentials, such as they were. He appeared tofindthereplypartiallyreassuring,andwentonto moretacticalquestions. WhomshouldIhavechosen ascaptainforthelasttestmatchayearbefore(inold- stylenums1930)? IftheselectorshaddecidedthatSnow wasthemantosaveEngland,whatwouldhavebeen 2 mystrategyandtactics? (‘Youareallowedtoact,ifyou aresufficientlymodest,asnon-playingcaptain.’) And soon,oblivioustotherestofthetable. Hewasquite absorbed. As I had plenty of opportunities to realize in the future,Hardyhadnofaithinintuitionsorimpressions, hisownoranyoneelse’s. Theonlywaytoassesssome- one’sknowledge,inHardy’sview,wastoexaminehim. Thatwentformathematics,literature,philosophy,poli- tics,anythingyoulike. Ifthemanhadbluffedandthen wiltedunderthequestions,thatwashislookout. First things came first, in that brilliant and concentrated mind. That night in the combination-room, it was nec- essary to discover whether I should be tolerable as a cricket companion. Nothing else mattered. In the end he smiled with immense charm, with child-like openness,andsaidthatFenner’s(theuniversitycricket ground)nextseasonmightbebearableafterall,with theprospectofsomereasonableconversation. Thus, just as I owed my acquaintanceship with LloydGeorgetohispassionforphrenology,Iowedmy friendshipwithHardytohavingwastedadispropor- tionateamountofmyyouthoncricket. Idon’tknow whatthemoralis. Butitwasamajorpieceofluckfor me. This was intellectually the most valuable friend- shipofmylife. Hismind,asIhavejustmentioned,was brilliantandconcentrated: somuchsothatbyhisside anyoneelse’sseemedalittlemuddy,alittlepedestrian and confused. He wasn’t a great genius, as Einstein and Rutherford were. He said, with his usual clarity, 3 thatifthewordmeantanythinghewasnotagenius atall. Athisbest,hesaid,hewasforashorttimethe fifthbestpuremathematicianintheworld. Sincehis characterwasasbeautifulandcandidashismind,he alwaysmadethepointthathisfriendandcollaborator Littlewoodwasanappreciablymorepowerfulmathe- maticianthanhewas,andthathisprotégéRamanujan reallyhadnaturalgeniusinthesense(thoughnottothe extent,andnothinglikesoeffectively)thatthegreatest mathematicianshadit. People sometimes thought he was under-rating himself,whenhespokeofthesefriends. Itistruethat hewasmagnanimous,asfarfromenvyasamancan be: butIthinkonemistakeshisqualityifonedoesn’t accept his judgment. I prefer to believe in his own statementinAMathematician’sApology,atthesame time so proud and so humble: ‘I still say to myself whenIamdepressedandfindmyselfforcedtolisten to pompous and tiresome people, “Well, I have done one thing you could never have done, and that is to havecollaboratedwithLittlewoodandRamanujanon somethinglikeequalterms.”’ Inanycase,hispreciserankingmustbelefttothe historiansofmathematics(thoughitwillbeanalmost impossiblejob,sincesomuchofhisbestworkwasdone incollaboration). Thereissomethingelse,though,at whichhewasclearlysuperiortoEinsteinorRutherford or any other great genius: and that is at turning any workoftheintellect,majororminororsheerplay,into aworkofart. Itwasthatgiftaboveall,Ithink,which made him, almost without realizing it, purvey such 4 intellectualdelight. WhenAMathematician’sApology wasfirstpublished,GrahamGreeneinareviewwrote thatalongwithHenryJames’snotebooks,thiswasthe bestaccountofwhatitwasliketobeacreativeartist. ThinkingabouttheeffectHardyhadonallthoseround him,Ibelievethatistheclue. He was born, in 1877, into a modest professional family. HisfatherwasBursarandArtMasteratCran- leigh,thenaminorpublic(Englishforprivate)school. His mother had been senior mistress at the Lincoln Training College for teachers. Both were gifted and mathematicallyinclined. Inhiscase,asinthatofmost mathematicians,thegenepooldoesn’tneedsearching for. Muchofhischildhood,unlikeEinstein’s,wastypi- calofafuturemathematician’s. Hewasdemonstrating aformidablyhighi.q. assoonas,orbefore,helearned totalk.Attheageoftwohewaswritingdownnumbers uptomillions(acommonsignofmathematicalability). When he was taken to church he amused himself by factorizingthenumbersofthehymns: heplayedwith numbers from that time on, a habit which led to the touchingsceneatRamanujan’ssick-bed: thesceneis well known, but later on I shall not be able to resist repeatingit. It was an enlightened, cultivated, highly literate Victorianchildhood. Hisparentswereprobablyalittle obsessive,butalsoverykind. ChildhoodinsuchaVic- torianfamilywasasgentleatimeasanythingwecould provide,thoughprobablyintellectuallysomewhatmore exacting. Hiswasunusualinjusttworespects. Inthe firstplace,hesufferedfromanacuteself-consciousness 5 atanunusuallyearlyage,longbeforehewastwelve. Hisparentsknewhewasprodigiouslyclever,andso didhe. Hecametopofhisclassinallsubjects. But,as the result of coming top of his class, he had to go in frontoftheschooltoreceiveprizes: andthathecould not bear. Dining with me one night, he said that he deliberatelyusedtotrytogethisanswerswrongsoas to be spared this intolerable ordeal. His capacity for dissimulation,though,wasalwaysminimal: hegotthe prizesallthesame. Some of this self-consciousness wore off. He be- camecompetitive. AshesaysintheApology: ‘Idonot rememberhavingfelt,asaboy,anypassionformathe- matics,andsuchnotionsasImayhavehadofthecareer ofamathematicianwerefarfromnoble. Ithoughtof mathematicsintermsofexaminationsandscholarships: Iwantedtobeatotherboys,andthisseemedtobethe wayinwhichIcoulddosomostdecisively.’ Neverthe- less, he had to live with an over-delicate nature. He seemstohavebeenbornwiththreeskinstoofew. Un- likeEinstein,whohadtosubjugatehispowerfulego inthestudyoftheexternalworldbeforehecouldat- tainhismoralstature,Hardyhadtostrengthenanego whichwasn’tmuchprotected. Thisattimesinlaterlife madehimself-assertive(asEinsteinneverwas)when hehadtotakeamoralstand. Ontheotherhand,itgave himhisintrospectiveinsightandbeautifulcandor,so thathecouldspeakofhimselfwithabsolutesimplicity (asEinsteinnevercould). Ibelievethiscontradiction,ortension,inhistem- peramentwaslinkedwithacuriousticinhisbehavior. 6 Hewastheclassicalanti-narcissist. Hecouldnoten- dure having his photograph taken: so far as I know, thereareonlyfivesnapshotsinexistence. Hewould not have any looking glass in his rooms, not even a shavingmirror. Whenhewenttoahotel,hisfirstac- tionwastocoverallthelooking-glasseswithtowels. Thiswouldhavebeenoddenough,ifhisfacehadbeen likeagargoyle: superficiallyitmightseemodder,since allhislifehewasgood-lookingquiteoutoftheordi- nary. But, of course, narcissism and anti-narcissism havenothingtodowithlooksasoutsideobserverssee them. Thisbehaviorseemseccentric,andindeeditwas. Between him and Einstein, though, there was a dif- ference in kind. Those who spent much time with Einstein—suchasInfeld—foundhimgrowstranger,less likethemselves, thelongertheyknewhim. Iamcer- tainthatIshouldhavefeltthesame. WithHardythe opposite was true. His behavior was often different, bizarrely so, from ours: but it came to seem a kind ofsuperstructuresetuponanaturewhichwasn’tall that different from our own, except that it was more delicate,lesspadded,finer-nerved. The other unusual feature of his childhood was moremundane: butitmeanttheremovalofallpracti- calobstaclesthroughouthisentirecareer. Hardy,with his limpid honesty, would have been the last man to be finicky on this matter. He knew what privilege meant,andheknewthathehadpossessedit. Hisfam- ilyhadnomoney,onlyaschoolmaster’sincome,but theywereintouchwiththebesteducationaladviceof 7
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