Adventures of a Mathematician - PDF Free Download (2025)

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AdventuresofaMathematician S.M.Ulam Prefacetothe1991EditionbyWilliamG.MathewsandDanielO.Hirsch NoteonS.M.Ulam'sMathematicsbyJanMycielski PostscriptbyFrançoiseUlam UniversityofCaliforniaPress Berkeley•LosAngeles•London

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Inmemoryofmyparents UniversityofCaliforniaPress BerkeleyandLosAngeles UniversityofCaliforniaPress London,England Copyright©1991by TheRegentsoftheUniversityofCalifornia Prefacecopyright©1983S.M.Ulam Copyright©1976S.M.Ulam LibraryofCongressCataloginginPublicationData Ulam,StanislawM. Adventuresofamathematician. Bibliography:p. Includesindex. 1.Ulam,StanislawM.I.Title. QA29.U4A33 75­20133 ISBN0­520­07154­9 3456789 PrintedintheUnitedStatesofAmerica ThepaperusedinthispublicationmeetstheminimumrequirementsofAmericanNationalStandardforInformationSciences—PermanenceofPaperforPrinted LibraryMaterials,ANSIZ39.48—1984

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CONTENTS Prefacetothe1991edition byWilliamG.MathewsandDanielHirsch

vii

Prefacetothe1983edition

xxi xxviii

Acknowledgments

xxix

ANoteonS.M.Ulam'sMathematics byJanMycielski

3

Prologue PartI:BecomingaMathematicianinPoland 1.Childhood

9

2.StudentYears

25

3.TravelsAbroad

52

PartII:AWorkingMathematicianinAmerica

4.PrincetonDays

65

5.HarvardYears

84

6.TransitionandCrisis

106

7.TheUniversityofWisconsin

123

PartIII:LifeamongthePhysicists

8.LosAlamos

141

9.SouthernCalifornia

172

10.BackatLosAlamos

188

11.The"Super"

209

12.TheDeathofTwoPioneers

225

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PartIV:ThePastFifteenYears

13.GovernmentScience

249

14.ProfessorAgain

266

15.RandomReflections

273

PostscripttoAdventures byFrançoiseUlam

305

Bibliography

319

Index

321

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PREFACETOTHE1991EDITION Itisstillanunendingsourceofsurpriseformetoseehowafewscribblesonablackboardoronasheetofpapercouldchangethecourseofhumanaffairs.

ThisremarkofStanislawUlam'sisparticularlyappropriatetohisowncareer.OurworldisverydifferenttodaybecauseofUlam'scontributionsinmathematics, physics,computerscience,andthedesignofnuclearweapons. WhilestillaschoolboyinLwów,thenacityinPoland,hesignedhisnotebook"S.Ulam,astronomer,physicistandmathematician."Oftheseearlyinterestsperhapsit wasnaturalthatthetalentedyoungUlamwouldeventuallybeattractedtomathematics;itisinthissciencethatPolandhasmadeitsmostdistinguishedintellectual contributionsinthiscentury.UlamwasfortunatetohavebeenbornintoawealthyJewishfamilyoflawyers,businessmen,andbankerswhoprovidedthenecessary resourcesforhimtofollowhisintellectualinstinctsandhisearlytalentformathematics.EventuallyUlamgraduatedwithadoctorateinpuremathematicsfromthe PolytechnicInstituteatLwówin1933.AsUlamnotes,theaestheticappealofpuremathematicsliesnotmerelyintherigorouslogicoftheproofsandtheorems,but alsointhepoeticeleganceandeconomyinarticulatingeachstepinamathematicalpresentation.Thisveryfundamentalandaristocraticformofmathematicswasthe concernoftheschoolofPolishmathematiciansinLwówduringUlam'searlyyears. ThepuremathematiciansatthePolytechnicInstitutewerenotsolitaryacademicrecluses;theydiscussedanddefendedtheirtheoremspracticallyeverydayinthe coffeehousesandtearoomsofLwów.Thisdeeplycommittedcommunityofmathematicians,inpursuingtheirworkthroughcollectivediscussioninpublic,allowed talentedyoungstudentslikeUlamtoobservetheintellectualexcitementandcreativityofpuremathematics.EventuallyyoungUlamcouldparticipateonanequal footingwithsomeofthemostdistinguishedmathemati­

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ciansofhisday.ThelongsessionsatthecafeswithStefanBanach,KazimirKuratowski,StanislawMazur,HugoSteinhaus,andotherssetthetoneofUlam'shighly verbalandcollaborativestyleearlyon.Ulam'searlymathematicalworkfromthisperiodwasinsettheory,topology,grouptheory,andmeasure.Hisexperiencewith thelivelyschoolofmathematicsinLwówestablishedUlam'slifelong,highlycreativequestfornewmathematicalandscientificproblems. AsconditionsinprewarPolanddeteriorated,UlamwelcomedopportunitiestovisitPrincetonandHarvard,eventuallyacceptingafacultypositionattheUniversityof Wisconsin.AsUnitedStatesinvolvementinWorldWarIIdeepened,Ulam'sstudentsandprofessionalcolleaguesbegantodisappearintosecretgovernment laboratories.FollowingafailedattempttocontributetotheAlliedwareffortbyenlistingintheU.S.military,UlamwasinvitedtoLosAlamosbyhisfriendJohnvon Neumann,oneofthemostinfluentialmathematiciansofthetwentiethcentury.ItwasatLosAlamosthatUlam'sscientificinterestsunderwentametamorphosisand wherehemadesomeofhismostfar­reachingcontributions. OnhisveryfirstdayatLosAlamoshewasaskedtoworkwithEdwardTeller'sgrouponthe"Super"bombproject,anearlyattempttodesignathermonuclearor hydrogenbomb.ExceptforTeller'ssmallgroup,thescientistsatLosAlamoswereworkingonthedesignandconstructionofanatomicbombbasedontheenergy releasedbythefissionorbreakupofuraniumorplutoniumnuclei.AlthoughtherewasageneralconsensusatLosAlamosthatthefissionbombwouldhavetoprecede theSuperforwhichitwouldserveasanignitionde­vice,TellerwasalreadypreoccupiedwiththeSuperandre­fusedtoworkonthefissionbombcalculations.Asa meansofretainingTelleratLosAlamos,RobertOppenheimeraslabdirectorallowedTellertoworkontheSuperbombwithseveralscientistsandassistants.Teller's assignmentforUlamonhisarrivalatLosAlamoswastostudytheexchangeofenergybetweenfreeelectronsandradiationintheextremelyhotgasanticipatedin thermonuclearbombs.Ironically,thisfirst­dayproblemforUlamin1943wouldlaterbecomeacriticalpart

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ofUlam'sworkwithCorneliusEverettin1950inwhichhedemonstratedthatTeller'sdesignfortheSuperbombwasimpractical. ThisfirstproblemintheoreticalphysicswasthebeginningofamajorscientifictransitionforUlamfromtheesoteric,abstractworldofpuremathematicstoaquite differentkindofappliedmathematicsnecessarytovisualizeandsolveproblemsinphysics.ThemathematicsrelevanttothephysicalproblemsatLosAlamosinvolved differentialandintegralequationsthatdescribethemotionofgas,radiation,andparticles.Thetransitionfrompuremathematicstophysicsisseldomattemptedand veryrarelyaccomplishedatUlam'slevel.Thecreativeprocessandtheinitialguessworkthatleadtosignificantnewideasinphysicsinvolveanaddeddimensionof tasteandjudgmentextendingbeyondtherigorouslogicofmathematicsalone.Physicalintuitionwhich"veryfewmathematiciansseemtopossesstoanygreatdegree" isconstrainedbyknowledgeofnaturalphenomenadeterminedfromexperiment.Ulamclaimsnottohaveexperiencedthis"gapbetweenthemodeofthinkinginpure mathematicsandthethinkinginphysics."Indeed,inthesememoirsUlamdiscusseshistransitionfrompuremathematicstomathematicalphysicsandhopesthathis analysis''ofthinkinginscienceisoneofthepossibleinterestsofthisbook." Ulamcouldhardlyhavebeeninbettercompanytolearnphysics.DuringthewaryearsthescientistsassembledatLosAlamosrepresentedaWho'sWhoofmodern physicalscience.Thelargenumberofeminentphysicists—HansBethe,NielsBohr,EnricoFermi,RichardFeynman,ErnestLawrence,J.RobertOppenheimer,and soon—formedanintellectualpowerhouseofphysicsthathasnotbeensurpassedbeforeorsince. DuringthewaryearsUlamcontributedtothedevelopmentofthefissionbombwithstatisticalstudiesonthebranchingandmultiplicationofneutronsresponsiblefor initiatingandsustainingthechainreactionandenergyreleaseinuraniumorplutonium.AcriticalproblemonwhichUlamworkedwithvonNeumannwasthedetailed calculationofthe

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implosionorcompressionofasphereofuraniumeffectedbyanexternalchemicaldetonation.Whenuraniumiscompressedthesmallnumberofnaturallyoccuring neutronscreatedbyrandomfissionsofuraniumnucleicollidemoreeasilywithotheruraniumnuclei.Someofthesecollisionsresultinfurtherfissions,multiplyingfurther thenumberofneutronsuntilarapidchainreactionensues,ultimatelyreleasinganextraordinaryamountofenergyinapowerfulexplosion.Inordertopredictthe amountofenergyreleased,LosAlamosscientistsneededtoestimatethedetailedbehavioroftheuraniumasitwasbeingcompressed.Althoughthisproblemwas conceptuallystraightforward,accuratesolutionswerenotpossibleusingstandardmathematicalanalyses.Thisproblemwasquiteliterallyatthesecretcoreofatomic bombresearchatLosAlamos—eventheword"implosion"wasclassifiedduringthewar. ButUlam'smostremarkableachievementatLosAlamoswashiscontributiontothepostwardevelomentofthethermonuclearorhydrogenbombinwhichnuclear energyisreleasedwhentwohydrogenordeuteriumnucleifusetogether.UlamwasaparticipantataLosAlamosmeetinginApril,1946,atwhichthewartimeefforts ontheSuperbombwerediscussedandevaluated.Theconceptualideaofthe"ClassicalSuper"wastoheatandignitesomepartofaquantityofliquiddeuteriumby usinganatomicbomb.Thethermalenergydepositedinthispartwouldinitiatedeuteriumreactionswhichwouldinturnheatadjacentregions,inducingfurther thermonuclearreactions,untilthedetonationwouldpropagatethroughtheentireamountofdeuteriumfuel.Deuterium,aheavierosotopeofhydrogenhavinganextra neutroninitsnucleus,waspreferredsinceitreactsatsignificantlylowertemperaturesthanordinaryhydrogen.Tritum,athirdandevenheavierformofhydrogenwith twoneutrons,reactsatevenlowertemperaturesbut,unlikedeuterium,isvirtuallynonexistentinnatureandwasextremelyexpensivetomakeinnuclearreactors. TheevaluationofTeller'sSuperprojectatthe1946meetingwasguardedlyoptimistic,buttheparticipantswereaware

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ofmajortechnicaluncertaintiesandpotentialdifficultieswiththeSuperdesign.IndiscussingtheconclusionsreachedatthemeetingJ.CarsonMarkhaswritten,"The estimatesavailableofthebehaviorofthevariousstepsandlinksinthesortofdeviceconsideredwereratherqualitativeandopentoquestionindetail.Themain questionofwhethertherewasaspecificdesignofthattypewhichwouldworkwellwasnotanswered."Studiespriorto1946hadestablishedthatthenetbalanceof energygainsoverlossesintheSuperbombwasmarginal;therewasnolargemarginofdesignflexibilityforwhichasuccessfuldetonationcouldbeguaranteed. AccordingtoMark, Asitwas,thestudiesofthisquestionhadmerelysufficedtoshowthattheproblemwasverydifficultindeed;thatthemechanismsbywhichenergywouldbecreatedinthe systemanduselesslylostfromitwerecomparable;andthatbecauseofthegreatcomplexityandvarietyofprocesseswhichwereimportant,itwouldrequireoneofthemost difficultandextensivemathematicalanalyseswhichhadeverbeencontemplatedtoresolvethequestion—withnocertaintythatevensuchanattemptcouldsucceedinbeing conclusive.

TheuncertaintiesregardingtheignitionandsustenanceoffusionreactionsintheSuperbombdesignasdevelopedbyTeller'sgroupduringthewaryearswerestill presentinlate1949andearly1950.Nevertheless,thiswasthehydrogenbombdesignthatTellerlobbiedforinWashingtonandthatformedthebasisofPresident Truman'sdecisionearlyin1950toaccelerateworkonthefusionbomb. ThetwomainquestionsabouttheSuperdesignwere(1)whetheritwouldbepossibletoignitesomeofthedeuteriumtogetthethermonuclearreactionsstarted,and (2)whetherathermonulcearreactionintheliquiddeuterium,oncestarted,wouldbeself­sustainingor,alternatively,wouldslowdownandfizzleawayiftheratethat energyislostfromthereactingregionsexceededthatproducedbythereactions.TheignitionoftheSuperwouldrequireagun­typeatomicbombtriggerinwhichtwo subcriticalmassesoffissionableuraniumwouldberapidlyunitedtoformasupercriticalex­

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plosivemass,asintheHiroshimabomb.Theignitionproblemwasdifficult.TheunusuallyhightemperaturesrequiredforignitionwouldrequireatriggerA­bombthat wouldneedtohaveayield,reachtemperatures,anduseaquantityoffissionablematerialsubstantiallyinexcessofthebombsinthearsenalin1950.Evenunderthe mostfavorablecircumstances,thedeuteriumcouldnotbeigniteddirectly.Itwasthoughtthatasmallamountoftritiumcouldbeusedtohelpinitiatedeuterium­burning intheregioninitiallyheatedbythefissionbomb. ThefirstamajordifficultyfortheSuper,theproblemofignition,wasattackedbyUlamonhisowninitiativebutincollaborationwithCorneliusEverett,amathematical colleagueofUlam'sattheUniversityofWisconsinwhohadcometoLosAlamosafterthewaratUlam'sinvitation.Thesecalculationsfollowedindetailtheinitial evolutionofthenuclearreactionsintritiumanddeuteriumandincludedanestimateoftheheatingoftheunburntnuclearfuelbythehotreactingregionswithallowances madefortheenergylostduetoexpansionandradiation.TheUlam­Everettcalculationwastediusandexacting.Whileeachstepofthecomputationwasunderstood, thecomplexinterplayamongthemanycomponentsinvolvedmadethewholecalculationextremelydifficult.Theexchangeofenergybetweenelectronsandradiation. Ulam'sfirstproblematLosAlamos,wasjustonepartofthismonumentalcalculation.ForseveralmonthsUlamandEverettworkedinconcentratedeffortfromfourto sixhoursaday.Sinceeachstepinthecalculationdependedonthepreviouswork,itwasnecessarytocompleteeachstagevirtuallywithouterror;fortunately, freedomfromerrorwasoneofEverett'sspecialties.Itishardtoimaginetodaythatthesecalculationswereperformedwithsliderulesandold­fashionedmanually operatedmechanicaldeskcalculators.UlamandEveretthadtomakemanyapproximationsandeducatedguessesinorderforasolutiontobepossibleatall.Bythis timeUlamhadclearlymasteredthephysicalintuitionandjudgmentneededtomakesensibleestimates.Whenthecalculationwasfinished,however,theirconclusion wasnegative.Thedeuteriumcouldnot

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beignitedwithoutspectacularamountsoftritum,amountssufficienttomaketheentireSuperprojectimpracticalanduneconomical.Withinafewmonthsthe conclusionsoftheUlam­EverettcalculationwereconfirmedbyvonNeumannusinganearlyelectroniccomputeratPrinceton. TheseconduncertaintyinthedesignoftheSuperwasthequestionofthepropagationofthedeuterium­burningregionthroughouttheentireamountofliquid deuterium.Wouldthefusionreactionbeself­sustainingassumingthattheignitiondifficultycouldsomehowbeovercome?Thisfundamentalproblemwassolvedby UlamincollaborationwiththebrilliantphysicistEnricoFermi.Againusingsliderulesanddeskcalculators—andgreatcareinmakingtheappropriatephysical approximations—theyreachedanothernegativeconclusion;theheatlostformthedeuteriumburningregionwastoogreattosustainthereaction.Indiscussingthe conclusionsoftheUlam­Fermicalculations,Ferminotedcautiouslythat"ifthecross­sectionsforthenuclearreactionscouldsomehowbetwoorthreetimeslarger thanwhatwasmeasuredandassumed,thereactioncouldbehavemoresuccessfully."Infactthecross­sections(whichcharacterizetheratethatthereactionscan occur)usedbybothTeller'sgroupandbyUlamandFermiin1950werelargerandthereforemoreotpimisticthanthemoreaccuratecross­sectionsobtained experimentallybyJamesTuckinthefollowingyear.InrecentyearstheUlam­Everettcalcualtionhasbeenredoneinamuchmorerefinedmannerusingmodern computersthathaveconfirmedthemarginalcharacteroftheself­sustainingpropagation. WithinmonthsofpresidentTruman'sdirectivetoexpeditethedevelopmentofathermonuclearbomb,thetwobasicassumptionsofTeller'sSupermodelwereshown byUlamandhiscolleaguestobeincorrect.Acrashprogramhadbegunonaprojectthatwasfundamentallyflawedandwhichhadneverbeenseriouslytestedpriorto Ulam'swork.AccordingtoHansBethe,Teller"wasblamedatLosAlamosforleadingtheLaboratory,andindeedthewholecountry,intoanadventurousprogramon thebasisofcalculationswhichhehimselfmusthaveknowntohavebeenveryincomplete."Theenergyre­

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leasedbythedeuteriumreactionwouldbelostbeforeadjacentregionscouldbeignitedsince,inUlam'sexplanation,"thehydrodynamicaldisassemblyproceeded fasterthanthebuildupandmaintenanceofthereaction."Teller,whohadworkedontheSuperduringthewaryearsandwholaterbecameaone­manpoliticalaction committeeurgingacrashprogramforitsconstruction,wasdistraughtandpracticallyundonebytheUlam­Everett­Fermiconclusions.Tellerhaswrittenthat"Ulam's workindicatedthatwewereonthewrongtrack,thatthehydrogenbombdesignwethoughtwouldworkbestwouldnotworkatall." ThecrisisofdisappointmentfollowingthesedevelopmentswasquitestunninglyresolvedbyUlaminFebruary,1951,whenhesuggestedameansofcompressingthe deuteriumsufficientlytoallowbothignitionandself­sustainingpropagation.AccordingtoHansBethe,directorofthetheoreticaldivisionatLosAlamosduringthewar, Ulam'sideawastouse"thepropagationof[a]mechanicalshock"(compression)wavefromafissionexplosiontoinduceastrongcompressioninthethermonuclear fuel,whichwouldsubsequentlyexplodewithgreatviolence.Theadvantagesofcompressioninhelpingtomakethermonuclearreactionsmoreefficienthadbeen discussedevenasearlyastheApril1946meeting,butwerenevertakenseriouslysincethecompressionrequiredwasfargreaterthancouldbeachievedwith chemicalexplosions.WhenUlamtoldTellerofhisideaofusingafissionbombtocompressthedeuteriumjustpriortoitsignition,Tellerimmediatelyperceivedthe valueoftheidea.However,Tellersuggestedthattheimplosioncouldbeachievedmoreconvenientlybytheactionofradiation,withaso­called"radiationimplosion," ratherthanwiththemechanicalshockproposedbyUlam(whichwouldalsohaveworked).Thenewideaforthehydrogenbomb,knowneuphemisticallyasthe "Teller­Ulamdevice,"wasrapidlyacceptedbyLosAlamosscientistsandgovernmentofficials.SincefirstproposedbyUlam,thecouplingofaprimaryfission explosionwithasecondaryfusionexplosionbymeansofimplosionhasbeenastandardfeatureofthermonuclearbombs.

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Allofthesedetailsconcerningtheoriginsofthehydrogenbomb,totheextentthatwecanputthemtogetherfromdeclassifiedinformation,underscoreUlamasfar moreinfluentialthanhaspreviouslybeenknown.NotonlywashethefirsttodismantletheearlierSuperconceptwhichhadbeensoinflexiblyproposedformany years,heprovidedthekeyideathatresolvedthedifficultiesofbothignitionandpropagation.Inthisinstance,morethananyotherinUlam'sscientificcareer,he demonstrated"howafewscribblesonablackboardoronasheetofpaper"havequiteradicallyandirreversiblychangedthecourseofhumanaffairs." Inviewoftheimpactthatthearsenalofnuclearweaponshashadonworldaffairs,itisintriguingthatUlamreturnsinhisautobiographyseveraltimestodiscussthe mindsetandsocialroleofweaponsscientistswhosequesterthemselvesintopsecretlaboratoriestoinventandconstructinstrumentsofpotentialmassdestruction. MostofthescientistswhoworkedatLosAlamosduringWorldWarIIwereshockedbytheannihilationofJapanesecitiesandelectedtoreturntoacademiclifeafter thewar.ItislikelythatmanyofthosewhostayedonatLosAlamosorreturnedlaterwereinherentlyapoliticaland,likeUlam,were"mainlyinterestedinthescientific aspectsofthework,"having"noqualmsaboutreturningtothelaboratorytocontributetofurtherstudiesofthedevelopmentofatomicbombs."AlthoughUlamlater feltthatthestockpileofnuclearweaponshadgrownlargerthannecessaryinhisviewtherewasnothingintrinsically"bad"aboutthemathematicsorthelawsofnature usedincreatingnewweapons.Knowledgeitselfiswithoutmoralcontent.Inparticular,Ulam''neverhadanyquestionsaboutdoingpurelytheoreticalwork"on nuclearweapons,leavingtootherstheirconstructionandapplicationtopoliticalandmilitaryends. Ulammakesacuriousdistinctionbetweentheacquisitionofknowledgeconcerningnewinstrumentsofmassdestructionbyscientistsanditswiderdissemination:"I sincerelyfeltitwassafertokeepthesemattersinthehandsofscientistsandpeoplewhoareaccustomedtoobjectivejudgmentsratherthaninthoseofdemagoguesor jingoists,orevenwell­meaning

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buttechnicallyuninformedpoliticians."However,inagovernment­fundedlaboratorysuchasLosAlamos,thesymbiosisthatexistsbetweenweaponstechnologyand politicaldecisionsisinescapable.WhileUlaminsiststhat"oneshouldnotinitiateprojectsleadingtopossiblyhorribleends,"itwouldneverthelessbe"unwiseforthe scientiststoturnawayfromproblemsoftechnology"since''thiscouldleaveitinthehandsofdangerousandfanaticalreactionaries."Inspiteoftheseapparent contradictions,Ulam'sjustificationsofhisroleinweaponsdevelopmentprovideuswithoneofthefewinsightsintothepersonalattitudesofaLosAlamosscientist towardtheendproductsofhiswork. ByvirtueofhisdefenseworkattheLosAlamosLaboratory,Ulamenjoyedmanyadvantagesnotavailabletoacademicscientists.Chiefamongthesewashisearly accesstothemostpowerfulandfastestcomputersinexistence.Forseveraldecadesafterthewar,thecomputingfacilitiesatthenationalweaponslabsfarexceeded thoseavailabletouniversityscientistsworkingonnon­classifiedresearch.ThiswasanadvantagethatUlamexploitedinavarietyofremarkableways. Thegrowthofpowerfulcomputerswasinitiallydrivenbythewareffort.AtthebeginningofWorldWarIItherewerenoelectroniccomputersinthemodernsense, onlyafewelectromechanicalrelaymachines.Duringthewar,scientistsattheUniversityofPennsylvaniaandattheAberdeenProvingGroundinMarylanddeveloped theENIAC,theElectronicNumericalIntegratorandComputer,whichhadcircuitryspecificallydesignedforcomputingartilleryfiringtablesfortheArmy.Bymodern standards,thisearlycomputerwasextremelyslowandelephantine:theENIACoperatingattheUniversityofPennsylvaniain1945weighedthirtytonsandcontained abouteighteenthousandvacuumtubeswith500,000solderedconnections.WhileonavisittotheUniversityofPennsylvaniain1944,JohnvonNeumannwas inspiredtodesignanelectroniccomputerthatcouldbeprogrammedinthemodernsense,onewhichcouldbeinstructedtoperformanycalculationandwouldnotbe restrictedtocomputingartillerytables.Thenewcomputerwouldhavecircuitsthat

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couldperformsequencesoffundamentalarithmeticoperationssuchasadditionandmultiplication.VonNeumanndesiredamoreflexiblecomputertosolvethe mathematicallydifficultA­bombimplosionproblembeingdiscussedatLosAlamos.ThefirstelectroniccomputeratLosAlamos,however,knownastheMANIAC (MathematicalAnalyzer,NumericalIntegratorandComputer),wasnotavailableuntil1952. OneofUlam'searlyinsightswastousethefastcomputersatLosAlamostosolveawidevarietyofproblemsinastatisticalmannerusingrandomnumbers,amethod whichhasbecomeappropriatelyknownastheMonteCarlomethod.ItoccurredtoUlamduringagameofsolitairethattheprobabilityofvariousoutcomesofthe cardgamecouldbedeterminedbyprogrammingacomputertosimulatealargenumberofgames.Newlyselectedcardscouldbechosenfromtheremainingdeckat random,butweightedbytheprobabilitythatsuchacardwouldbethenextselected.Thecomputerwoulduserandomnumberswheneveranunbiasedchoicewas necessary.Whenthecomputerhadplayedthousandsofgames,theprobabiltyofwinningcouldbeaccuratelydetermined.Inprincipletheprobabilityofsolitaire successcouldberigorouslycalculatedusingprobabiltytheoryratherthancomputers.However,thisapproachisimpossibleinpracticesinceitwouldinvolvetoomany mathematicalstepsandexceedinglylargenumbers.TheadvantageoftheMonteCarlomethodisthatthecomputercanbeefficientlyprogrammedtoexecuteeach stepinaparticulargameaccordingtoknownprobabilitiesandthefinaloutcomecanbedeterminedtoanydesiredprecisiondependingonthenumberofsample gamescomputed.ThegameofsolitaireisanexampleofhowtheMonteCarlomethodcanbeusedtosolveotherwiseintractableproblemswithbrutecomputational power. AnearlyapplicationoftheMonteCarlomethodusinghighspeedcomputerswastostudythepropagationofneutronsinfissionbombs.Thiswasaccomplishedby randomlypickingthepositionofaradioactivenucleusthatwouldreleaseaneutron,thenrandomlyselectingtheneutron'senergy,itsdirectionofmotion,andthe distancetheneutronwouldtravelbefore

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eitherescapingorcollidingwiththenucleusofanotheratom.Inthelatterevent,theneutronwouldeitherbescattered,absorbed,orcouldinducenuclearfission accordingtoprobabilitiesagainselectedwithrandomnumbers.Inthismanner,aftermanyneutronlifeexperienceshadbeencalculated,itwaspossibletodetermine thenumberofneutronsatanyenergymovinginaparticulardirectionatanypositionintheapparatus.TheMonteCarlomethodisalsowell­suitedtocomputingthe equilibriumpropertiesofmaterials,inestimatingtheefficiencyofradiationorparticledetectorshavingcomplicatedgeometries,andinsimulatingexperimentaldatafor awidevarietyofphysicalproblems. AnotherearlyuseofcomputertechnologyinwhichUlammadecontributionsistheproblemofdeterminingthemotionofcompressiblematerial.Indeed,itwasthe calculationofimplodingcompressionwavesinthefissionablecoreofatomicbombsthatinitiallyattractedLosAlamosscientiststotheadvantagesoffastcomputers. OneofUlam'scontributionswashisideatorepresentthecompressiblematerialwithanensembleofrepresentativepointswhosemotioncouldbedeterminedbythe computer.Alongsimilarlines,Ulamperformedthefirststudiesofthesubtlycomplexcollectivemotionofstarsinacluster,eachmutuallyattractedtoalltheothersby gravitationalforces.TheapplicationsofcomputerstobothcompressiblematerialandstellarsystemsalonglinesfirstexploredbyUlamaremajorareasofresearch interesttoday. OfparticularinterestisUlam'sfarsightedcomputerexperimentinthemid­fiftieswithJohnPastaandEnricoFermiontheoscillationsofachainofsmallmasses connectedwithslightlynonlinearsprings.Anonlinearspringisonethatdoesnotquitestretchinexactproportiontotheamountofforceapplied.Whenthegroupof massessimulatedbythecomputerwasstartedoutinaparticularrathersimplemotion,Ulamandhiscolleaguesdiscoveredtotheiramazementthatthemasses eventuallyreturnednearlytotheoriginalmotionbutonlyafterhavinggonethroughabizarreandtotallyunanticipatedintermediateevolution.Todaycomputerstudies ofsuchnon­

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linearsystemshavebecomeamajorareaofinterdisciplinaryscientificinvestigations.Manystrangepropertiesofdynamicalsystemshavebeendiscoveredwhichhave ledtoadeeperunderstandingofthelong­termpropertiesofnonlinearsystemsobeyingdeceptivelysimplephysicallaws. ArelatedcomputerexperimentinspiredbyUlamwasthestudyofiterativenonlinearmappings.Thecomputerisprovidedwitha(nonlinear)rulefortransformingone pointinamathematicallydefinedregionofspaceintoanother,thenthesameruleisappliedtothenewpointandtheprocessiscontinuedformanyiterations.When examinedafteronlyafewiterations,thepatternisgenerallyuninteresting,butwhenacomputerisusedtogeneratethousandsofiterations,UlamandhiscolleaguePaul Steinobservedthatavarietyofstrangepatternscanresult.Insomecasesaftermanyiterationsthepointsconvergetoasinglepointorareorderedalongacurve withinthegivenregionofspace.Inothercasesthesuccessiveimagesofiteratedpointsappeartohavedisordered,chaoticproperties.Thefinalpatternofiterated imagescanbesensitivetotheinitialpointchoseningeneratingthemaswellastherulesfor(nonlinear)iteration.InrecentyearsthisearlyworkofUlamandSteinhas beengreatlyextendedatLosAlamos,nowamajorcenterofnonlinearstudies. Ulamalsohadaninterestintheapplicationofmathematicstobiology.Oneexamplethatmayhavebiologicalrelevanceisthesubfieldofcellularautomatafoundedby vonNeumannandUlam.Asanexampleofthisclassofproblemsimaginedividingaplaneintomanysmallsquareslikeacheckerboardwithseveralobjectsplacedin nearbysquares.Thenspecifyrulesfortheappearanceofnewobjects(orthedisappearanceofoldobjects)ineachsquaredependingonwhetheradjacentsquares areoccupiedornot.Witheachapplicationoftherulestoallthesquares,thepatternofoccupiedsquaresevolveswithtime.Dependingontheinitialconfigurationand therulesofgrowth,somecomputergeneratedcellularautomataevolveintopatternsresemblingcrystalsorsnowflakes,othersseemtohaveanever­changingmotion asiftheywerealive.Insome

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casescoloniesofself­replicatingpatternsexpandtofilltheavailablespacelikethegrowthofcoralorbacteriainapetridish. StanislawUlamwasamanofmanyideasandafertileimagination.HiscreativeandvisionarytalentplantedintellectualseedsfromLwówtoLosAlamoswhichhave flourishedintonewdisciplinesofstudythroughouttheworld.Ulam'sscientificworkwascharacterizedbyasingularlyverbalstyleofinquirybegunduringhisearly experiencesinthecoffeehousesofLwów.TheuseofwrittenmaterialwasalsolessessentialforUlamduetohisformidablememory—hewasabletorecitemany decadeslaterthenamesofhisclassmatesandtoquoteGreekandLatinpoetrylearnedasaschoolboy.Ulam'sverbalandsociallyinteractiveapproachwasinfact wellsuitedtotheresearchenvironmentatLosAlamos.TalentedcolleaguestherewereavailabletocollaboratewithUlam,toprovidethemissingdetailsoftheideas hesketchedout,andtopreparethescientificpapersandreportswhichchangedthecourseofhumanaffairs. WILLIAMG.MATHEWS DANIELO.HIRSCH

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PREFACETOTHE1983EDITION InwritingaprefacetoanothereditionofthisbookIcannotresistthetemptationtocomparethepresentwiththeguessesandtimidpredictionsImadeaboutthefuture ofscienceasitlookedtometenyearsago.Ifanything,thepresentlooksevenmoreexcitingthanIhadhoped.Itiswonderfultoobservehowmanyunforeseenor unforeseeablefactsandideashaveemerged.WhileIshallmentionjustafewofthemanydevelopmentsinrecentscience,itisimportanttorealizethattherateat whichwecomprehendtheuniverseisasvitalaswhatwefinallyunderstand. Progressinscienceandtechnologyhasproceededatanever­increasingpace,makingtheshortperiodsinceIwrotethisbookassignificantasanyinthehistoryof science.Toseethisonehasonlytothinkofthelandingsonthemoon,thenowcommonplacelaunchingofsatellites,andtheenormousdiscoveriesmadeinastronomy andinthestudyoftheearthitself. Mostnotablehasbeentheexponentialgrowthinthetechnologyofelectroniccomputers,whoseusepervadesmanyaspectsofdailylife.Nowelementsofa"meta­ theory"ofcomputingarebeingoutlinedandproblemsofcomputabilityinthegeneralsense,especiallywithrespecttoitslimits,arebeingstudiedsuccessfully. IwonderwhatJohnvonNeumann'sreactionwouldhavebeenhadhelivedtoseeitall.Heprophesiedthegrowingimportanceofthecomputer'srole,butevenhe wouldprobablyhavebeenamazedatthescopeofthecomputerageandtherapidityofitsappearance. Onecouldsaythataftertheatomicagetherecamethecomputerage,which,inturn,madethespaceagepossible.Allspacevehicles—rockets,satellites,projectiles, shuttles,andsoon—dependonthefeasibilityofveryfastcalculationsthatmustbeinstantlytransmittedtotheminouterspacetocorrecttheirorbits.Beforetheadvent ofthe

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fastestelectroniccomputersthiskindofremotecontrolwasnotpossible. Recentlyagreatwealthofobservationsinphysicsandastronomyhasincreasedtheperplexityofthedescriptionoftheuniverse.Theenigmaofquasarsisstill unresolved.Thesequasi­stellarobjectsseemtobebillionsoflight­yearsawaywithanintrinsicluminosityhundredsoftimesgreaterthanthatofthegalaxiesintheir foreground.InthefewyearssinceIwrotethisbook,vast"emptyregions"hundredsofmillionsoflight­yearswidehavebeenfound.Theseareasmakeusquestionthe samenessandisotropyoftheuniversesuggestedbytheapparentuniformityofthecosmicradiationremainingfromtheBigBang.Itisnowwidelybelievedthatblack holesdoexist.Theymayexplainthebehaviorofseveralobservedastronomicalobjects.Inaddition,growingevidencesupportsthetheorythatviolentprocesses causegiganticexplosionsinstarlikeobjectsandgalaxies. Toamathematicianlikemyself,thequestion,"Istheuniverseinspacefiniteandbounded,ordoesitextendindefinitely?"remainsthenumber­oneproblemof cosmogonyandcosmology. Inphysics,thenumberofnew,fundamental,orprimaryparticlesisconstantlyincreasing.Quarksseemmoreandmoretorepresentreal,notmerelymathematical, constituentsofmatter,buttheirnumberandnatureremainunverifiable,andscientistsareconsideringtheexistenceofsubparticles,suchasgluons. Sincethefirstpublicationofthisbookithasbecomemorelikely,itseemstome,thattheremightbeaninfinitechainofdescendingstructures.Toparaphraseawell­ knownstatementaboutfleas,largequarkshavebiggerquarksontheirbackstobitethem,bigoneshavebiggerones,andsoonadinfinitum. Thereisalsomuchspeculationabouttheidentityoforsimilaritybetweenthedifferentforcesofnature.Certainlythereisastronganalogybetweenelectromagnetic forcesandso­calledweakinteractions.Theremayevenbea

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mathematicalanalogybetweentheseforces,nuclearforces,andgravitationalforces. Mathematicsremainsthetoolforinvestigatingproblemssuchasthese.Electroniccomputershavehelpedimmenselyinsolvingcomplexcalculations,andagreatmany newresultshaveappearedinpuremathematicaldisciplinessuchasnumbertheory,algebra,andgeometry.Thebroadeningrangeof"constructive"mathematical methods,suchastheMonteCarlomethod,indicatesthatatheoryofcomplexitymaysoonaffectmanybranchesofmathematicsandstimulatenewpointsofview. Somephysicalproblemssuchasthestudyandinterpretationofparticlecollisiononthenew,miles­longacceleratorscallforgiganticMonteCarlomodeling. Presentlyinvogueisthestudyofnonlineartransformationsandoperations.ThesebeganintheLosAlamosLaboratory,whichnowhasaspecialcenterdevotedto nonlinearphenomena.Thiscenterrecentlyheldaninternationalconferenceonchaosandorder.Forthemostpartthisworkconcernsthebehaviorofiterations— repetitionsofagivenfunctionorflow.Theseproblemsrequireguidancefromwhatareessentiallymathematicalexperiments.Trialsonacomputercangivea mathematicianafeelingorintuitionofthequalitativebehavioroftransformations.SomeofthisworkcontinuesastudymentionedinChapter12,andsomefollows workPaulStein,I,andothershavedoneintheinterveningyears. Whilemuchofphysicscanbestudiedusinglinearequationsinaninfinitenumberofvariables(asinquantumtheory),manyproblems—hydrodynamicsincluded—are notlinear.Itisbecomingmoreandmorelikelythattheremaybenonlinearprinciplesinthefoundationsofphysics.AsEnricoFermioncesaid,"Itdoesnotsayinthe Biblethatalllawsofnatureareexpressiblelinearly!" ToanamateurphysicistsuchasIamtheincreasingmathematicalsophisticationoftheoreticalphysicsappearstobringaboutadecreaseintherealunderstandingof boththesmall­andthelarge­scaleuniverse.Theincreasing

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fragmentationmaybedueinparttoneglectintheteachingofthehistoryofscienceandcertainlytothegrowthofspecializationandoverspecializationinvarious branchesofscience,inmathematicsinparticular.AlthoughIamsupposedtobeafairlywell­readmathematician,therearenowhundredsofnewbookswhosevery titlesIdonotunderstand. Iwouldliketodevoteafewwordstowhatismanifestlytheageofbiology.Ibelievethesepastsixteenyearshaveseenmoresignificantadvancesinbiologythanin othersciences.Eachnewdiscoverybringswithitadifferentsetofsurprises.Genesthatweresupposedtobefixedandimmutablenowappeartomove.Theportion ofthecodedefiningagenemay"jump,"changingitslocationonthechromosome. Wenowknowthatsomesegmentsofthegeneticcodedonotexpressformulaeforthemanufactureofproteins.Thesesometimeslongishsequences,calledintrons,lie betweenchromosomesegmentsthatdocarryinstructions.Whatpurposeintronsserveisstillunclear. Thesuccessofgenesplicing—theinsertionorremovalofspecificgenesfromachromosome—hasopenedanewworldofexperimentation.Theapplicationofgene manipulationtosciencessuchasagriculture,forexample,mayhavealmostlimitlessbenefits.Inmedicinewecanalreadyproducehuman­typeinsulinfromgenetically alteredbacteria.Scientistshaveagreedtotakeprecautionsagainstaccidentallycreatingdangerousnewsubstancesingene­splicingexperiments.Thisseemstosatisfy theprofessionalbiologists.Still,thereisagreatdebateoverwhethertoallowunregulatedgeneticengineering,withallitspossibleconsequences. Myarticle"SomeIdeasandProspectsinBiomathematics"(seeBibliography)isanexampleofsomeofmyowntheoreticalworkinthisarea.Itconcernswaysof comparingDNAcodesforvariousspecificproteinsbyconsideringdistancesbetweenthem.Thisleadstosomeinterestingmathematicsthat,interalia,maybeused tooutlinepos­

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sibleshapesoftheevolutionarytreeoforganisms.TheideaofusingthedifferentcodesforacytochromeCwassuggestedandfirstinvestigatedbythebiologist EmanuelMargoliash. AtLosAlamos,agroupledbyGeorgeBell,WalterGoad,andotherbiologistsisusingcomputerstostudythevastnumberofDNAcodesnowexperimentally available.ThegroupwasrecentlyawardedacontractbytheNationalInstituteofHealthtoestablishalibraryofsuchcodesandtheirinterrelations. Itiswellknownthatgradualchanges,nomatterhowextensive,arebarelynoticeablewhiletheyoccur.Onlyafteracertainamountoftimedoesonebecomeawareof anytransformation.OnemorninginLosAlamosduringthewar,Iwasthinkingabouttheimperceptiblechangesinmyownlifeinthepastyearsthathadledtomy comingtothisstrangeplace.IwaslookingattheblueNewMexicoskywhereafewwhitecloudsweremovingslowly,seeminglyretainingtheirshape.WhenIlooked awayforaminuteandbackupagain,Inoticedthattheynowhadcompletelydifferentshapes.AcoupleofhourslaterIwasdiscussingthechangesinphysicaltheories withRichardFeynman.Suddenlyhesaid,"Itisreallyliketheshapeofclouds;asonewatchesthemtheydon'tseemtochange,butifyoulookbackaminutelater,itis allverydifferent."Itwasacuriouscoincidenceofthoughts. Changesarestilltakingplaceinmypersonallife.In1976IretiredfromtheUniversityofColoradotobecomeprofessoremeritus,asoberingtitle.Atthesametime,I acceptedapositionasresearchprofessorattheUniversityofFloridainGainesville,whereIstillspendafewmonthseveryyear,mostlyduringthewinterwhenitisnot toohot. Mywife,Françoise,andIsoldourBoulderhouseandboughtanotheroneinSantaFe,whichhasbecomeourbase.FromSantaFeIcommutethreeorfourtimesa weektotheLosAlamosLaboratory.Itssuperbscientificlibraryandcomputingfacilitiesallowmetocontinueworkinginsomeoftheareasof'sciencementioned above.Françoise

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actsasmy''HomeSecretary,"asIcallher,alludingtothetitleoftheBritishInteriorSecretary.WestilltravelquiteextensivelyandIcontinuetolectureinvarious places. WearefortunatethatourdaughterClairealsolivesinSantaFewithherhusband,StevenWeiner,anorthopedicsurgeon.Theirdaughter,nowfive,givesmeoccasion towonderathowremarkablethelearningprocessesofsmallchildrenare,howachildlearnstospeakandusephrasesanalogoustoandyetdifferentfromtheonesit hasheard.ObservingRebeccaspeakprovidesmewithadditionalimpulsesandexamplesfordescribingamathematicalschemaforanalogyingeneral. Mycollaborator,DanMauldin,aprofessoratNorthTexasStateUniversity,hasrecentlyeditedanEnglishversionofTheScottishBookmentionedinChapter2.We arenowcollaboratingonacollectionofnewunsolvedproblems.ThisbookwillhaveadifferentemphasisfromthatofmyCollectionofMathematicalProblems, publishedin1960.Thenewcollectionwilldealmorewithmathematicalideasconnectedtotheoreticalphysicsandbiologicalschemata. Manyofthepeoplementionedinthisbookhavesincedied,orleft,asmyfriendPaulErdöspreferstosay:KazimirKuratowski,myformerprofessor;KarolBorsuk andStanislawMazur,myPolishcolleagues;mycousinsJulekUlaminParisandMarysiaHarcourt­Smith;inBoulder,JaneRichtmyer,whohelpedwiththefirstwriting ofthisbook;GeorgeGamowandhiswifeBarbara;mycollaboratorsJohnPastaandEdCashwelloftheMonteCarloexperiments;andhereinLosAlamos(withina fewmonthsofeachother)theBritishphysicistJimTuckandhiswifeElsie.AsHoracesaid,"Omneseademidimur,omniumversatururna...sorsexitura...." AfewweeksagoIwasinvitedtogiveaSundaytalkattheLosAlamosUnitarianChurchonthesubjectof"PureScienceinLosAlamos."Thediscussionthat followedcenteredonproblemsthatareofgrowingconcernnowadays:therelationofsciencetomorality;thegoodandthebadin

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scientificdiscoveries.Around1910,HenriPoincaré,thefamousFrenchmathematician,hadconsideredsuchdilemmasinhisDernièresPensées.Thequestionswere lessdisturbingthen.Now,thereleaseofnuclearenergyandthepossibilityof'genemanipulationhavecomplicatedtheproblemsenormously. IwasaskedwhatwouldhavehappenedhadtheLosAlamosstudiesprovedthatitwasimpossibletobuildanatomicbomb.Theworld,ofcourse,wouldbealess dangerousplaceinwhichtolive,withouttheriskofsuicidalwarandtotalannihilation.Unfortunately,proofsofimpossibilityarealmostnonexistentinphysics.In mathematics,onthecontrary,theyprovidesomeofthemostbeautifulexamplesofpurelogic.(ThinkoftheGreeks'proofthatthesquarerootoftwocannotbea rationalnumber,thequotientoftwointegers!)Humanity,itseems,isnotemotionallyormentallyreadytodealwiththeseenormousincreasesinknowledge,whether theyinvolvethemasteryofenergysourcesortheinanimateandprimitivelifeprocesses. Someoneintheaudiencewonderedifsomeofthecurrentresearchonthehumanbrainmightnotultimatelyleadtoawiserandbetterworld.Iwouldliketothinkso, butthispossibilityliestoofarinthefuturetoevenguessat. Intheshortspanofmylifegreatchangeshavetakenplaceinthesciences.Seventyyearsamountstosome2percentofthetotalrecordedhistoryofmankind.I mentionedthisoncetoRobertOppenheimeratPrinceton.Hereplied,"Ah!butone­fiftiethisreallyalargenumber,excepttomathematicians!" SometimesIfeelthatamorerationalexplanationforallthathashappenedduringmylifetimeisthatIamstillonlythirteenyearsold,readingJulesVerneorH.G. Wells,andhavefallenasleep. S.M.U. SANTAFE AUGUST1982

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ACKNOWLEDGEMENTS Thisbookcouldhavebeenwrittenwithoutthehelpofmywife,Françoise,butitwouldhavebeenmerelyachaoticassemblageofitems.Thatitmaypresentsome coherentfeaturesistheresultofherinterventionandcollaboration.Shemanagedtodecreasesubstantiallytheentropyofthiscollectionofreminiscencesthrough severalyears'intelligentandsystematicwork.ThanksarealsoduetoGian­CarloRota,forournumerousconversationsonsomeofthetopicsofthisbook;toMrs. EmiliaMycielska,forherresearchonmydeceasedPolishcolleagues;andtoMrs.JaneRichtmyerforgoingoversomeoftherougherspotsofthetext. Gratefulacknowledgmentforpermissiontoreproducephotographsismadetothefollowing: TheSocietyofFellows,Harvard(HarvardJuniorFellows,1938) LosAlamosScientificLaboratory(allphotographssocreditedinthecaptions) HaroldAgnew(EnricoFermiinthe1940s) NicholasMetropolis(VonNeumann,Feynman,andUlamatBandelierLodge) TheVikingPress(GeorgeGamow'scartoonofthe"super"committee).FromMyWorldLinebyGeorgeGamow;copyright©1970byGeorgeGamow. LloydShearer(StanandFrançoiseUlamathome,1964) Allpicturesnototherwisecreditedarethepropertyoftheauthor.

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ANOTEONS.M.ULAM'SMATHEMATICS StanislawUlambelongedtoagroupofmathematicianswhocametotheUnitedStatesfromPolandbeforeandduringtheSecondWorldWarandplayedavery importantroleinthelifeofmathematicsinthiscountry.AmongthemonecannameNatanAronszajn,StefanBergman,SamuelEilenberg,WitoldHurewicz,Mark Kac,OttonNikodym,AlfredTarski,andAntoniZygmund.Theyescapedthefateofmanymillionskilledduringthewar,amongwhomweretheimportantPolish mathematiciansJozefMarcinkiewicz,StanislawSaks,andJuliuszSchauder.Thehistoryofmathematicswouldhavebeenverydifferentifthewarhadnotdecimated thePolishyouthandalmostexterminatedtheJewishyouthofEurope. Ulamwroteover150technicalpapersandthreebooks:ACollectionofMathematicalProblems,publishedin1960,MathematicsandLogic(withMarkKac), publishedin1968,andthisautobiographypublishedin1976.SelectionsofhispaperswerepublishedinSets,Numbers,Universes*(referredtoasSNU),Science, Computers,andPeople**andAnalogiesBetweenAnalogies***(referredtoasABA).AcollectionofarticlesabouthiscontributionsappearedinFrom CardinalstoChaos.**** Françoise,hiswife,hadagreatinfluenceuponhislife.Shemanagedtheirvarioushomeswithincredibleefficiencyandhelpedintheorganizationofhismanytravels.In additionsheeditedthisautobiographyfromtapesandwasacoeditorofthecollectionsofhispaperspublishedposthumously. Endowedwithexceptionalcharmandintelligence,Ulamhadanunusualfacilityforestablishingcontactswithpeople.Hismostpleasantqualitywashisopennessand spontaneity.Withouthesitationhesharedhisideaswithallwhowereinterested.Moreoverhehadaremarkablememoryandabroad *MITPress,1974. **Birkhäuser,Boston,1986. ***UniversityofCalforniaPress,1990. ****CambridgeUniversityPress,1989.

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humanisticcultureacquiredinanexcellentsecondaryschoolinLwów. Ulamhadauniqueabilitytoraiseimportantunsolvedproblems.Theseproblemsexertedanexceptionalinfluenceupontheworkofmanyscientists.Intheyearsofour acquaintancewhichbeganin1969,hepreferredtoinventopenquestions,especiallythoseattheboundaryofmathematics,physics,andbiology,ratherthantogointo thedetailsofmathematicalwork.Hebecameapioneerofcomputerapplicationstotheheuristicstudyofdynamicalsystems.Althoughheencouragedyounger mathematiciansandlikedtoseetheworkofpeoplefromitsbestside,helookedatmathematicsfromthepointofviewofascientistwhosepurposeistostudynature. Hecriticizedthesemathematicalproblemswhichdonotappeartohaveadirectornaturalmotivation,callingthem"Chinesepuzzles."Hepreferredtothinkabout physicsandbiology,andthemathematicalproblemsthatderivefromthem. ThisgiftisbestillustratedbyoneofhisconjectureswhichwassoonprovedbyK.Borsuk.Itiscalled"theantipodaltheorem."Toeachcontinuousfunctionffromthe sphericalsurfacetotheplane,thereexisttwoantipodalpointsonthesphere,xand­x,suchthatf(x)=f(­x). Awell­knownconsequenceoftheantipodaltheoremwasformulatedbySteinhausandiscalled"thehamandcheesesandwichtheorem."Itsaysthatifyouhavea sandwichcomposedofbread,ham,andcheese,thereisoneplanewhichevenlydividesthebread,theham,andthecheese.Inmoreformallanguage:Foranythree measurablesetsinthree­dimensionalspace,thereexistsaplanewhichhalveseachofthem. Ulam'sworkpertainstosomanyareasofmathematicsandothersciencesconnectedwithit,andissomanifold,thatonlyhismostimportantcontributionscanbe mentionedhere.

SetTheory Ulam'sdoctoraldissertation,writtenin1931(SNU,pp.9–19),pertainingtothesizeofcertaininfinitecardinalnumbers,

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containsresultswhichareveryimportantforthefoundationsofmodernmathematics. LikegeometryinthetimesofEuclid,modernmathematicscanbesynthesizedwithinacompactlistofaxioms.Theconceptsusedinthataxiomatizationbelongto Cantor'ssettheory.Strictlyspeakingwehaveaninfinitehierarchyofstrongerandstrongersettheories.Thefirstofthem,calledZermelo­Fraenkelsettheory(and denotedZF)sufficesforalmostallmathematics.Butanintriguingpartisstillmissing.Thetheoremsofthatpartrequiresomestrongertheoriesofthishierarchy.(Those theoremsbelongtodescriptivesettheory,problemsofmeasurability,capacitability,propertyofBaire,determinacy,thetheoryofidealsofsets,etc.)Themostnatural waytogaugethestrengthofthosetheoriesistolookatthesizeofinfinitecardinalnumberswhoseexistencecanbeprovedinthem. Ulam'sworkconstituesthefirstdeeperinvestigationofthesizeofafewsuchgauge­cardinals.Since1931thetheoryhasundergonesubstantialdevelopment.Butit wasnotuntil1960,almostthirtyyearsafterhiswork,thatW.HanfandA.Tarski,usingatheoremofJ.Los,madethenextsteponthatroad.Sincethenmany mathematicians(Keisler,Martin,Reinhardt,Solovay,andWoodin,tonameonlyafew),havedevelopedthistopic. UlamwasalsothefirsttodefineinTheScottishBook*thebinaryinfinitegameofperfectinformationwhichwasusedlaterbyH.Steinhausandthiswritertoexpress theaxiomofdeterminacy.(SeecommentaryinTheScottishBook,pp.113–116).Ulam'scontributionstosettheoryarereproducedandtheirsubsequentinfluenceis outlinedinSNU.

ErgodicTheoryandMeasureTheory BeforethewarSchreierandUlamwerestudyingthegroupofhomeomorphismsofthen­dimensionalsphereSninRn+1 *Birkhäuser,editedbyR.D.Mauldin,1979.

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(seeSNU).LaterOxtobyandUlam(SNU)provedseveralfundamentalresultsaboutthisgroup. Themostimportantofthesecanbeexplainedasfollows(omittingmathematicalprecision):Ifwegentlymixaglassofwater,wegetatransformationwhichis continuous(becausewaterhassomeviscosity)andpreservesvolumes(becausewaterisnearlyincompressible).Nowtheirtheoremtellsusthatalmostallsuch transformationshavethepropertythatnoportionofthewaterwhichhasapositivevolume,exceptthewholeglass,willoccupythesamelocationbeforeandafterthe mixing. Transformationswiththeabovepropertiesarecalledergodic,andbeforetheworkofOxtobyandUlam,itwasnotevenknownthatsuchtransformationsexist.

Topology Ulamhadaverygeometricalwayofthinkingandhewasinterestedinproblemsofgeometrictopology(e.g.,theantipodaltheoremmentionedabove).Buthealso contributedtogeneraltopology. WithKuratowskiheextendedFubini'sclassicaltheoremfromtheparadigmofmeasuretotheparadigmoftheBairecategory.WithSchreierheshowedthatevery homeomorphismofthesphereSncanbeuniformlyapproximatedbyappropriatecompositionsoffourfixedhomeomorphisms.WithBorsukheintroducedthe conceptsofsymmetricproductsandapproximatehomeomorphisms.WithAuerbachandSchreierheshowedthatinthesenseofmeasure,almosteverypairof elementsofasemi­simplecompactLiegroupgeneratesadensesubgroup.Allthesepapersgeneratedmanycontinuationsandexpansionsinthesubsequentliterature (seeSNUandcommentariestherein).

BranchingProcesses EverettandUlamwroteimportantpapersaboutprocessesoftheGalton­Watsontype(ABA).Theystudiedtheprobabilityconnectedwiththecascadesof elementaryparticlescausedbycollisionsofenergeticparticles.Theirchiefresultsper­

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tainedtotherelativenumberofparticlesofdifferentkindssuchasneutronsanduraniumnuclei.Sincethattime,thetheoryofbranchingprocesseshasundergonea greatdealoffurtherdevelopment.

NonlinearSystems ThetheoryofnonlinearsystemswasinitiatedbytheworkofFermi,Pasta,andUlam(SNU,ABA).Experimentingwithacomputer,theydiscoveredthatavibrating stringwhoseclassicalequationisperturbedbyacertainnonlineartermalmostreturnstoitsoriginalposition,andmuchearlierthanexpectedfromstatistical considerations.Thus,inspiteoftheexistenceofthenonlinearterm,energydoesnotdisperseoverhigherharmonicsbutremainsconcentratedoverthefirstfewand oftenreturnstotheveryfirst.Thisworkinitiatedagreatnumberofstudiesbyphysicistsandmathematicians;inparticularitledtothetheoryofsolitons. SteinandUlamobtainedinterestingresultsonnonlineartransformations,theirstableandnon­stablefixedpoints,periodicpoints,andothers(seeSNUorABA).The subjectoftheirstudiesisconnectedwiththeclassicalpaperofVolterraaboutthefluctuationofrelativepopulationnumbersofdifferentspeciesoffishesinponds.The relativenumberofindividualsofeachofthenspeciesisrepresentedbybaricentriccoordinatesofapointinann­1dimensionalsimplex.Thesimplexissubjecttoa transformationwhichdescribesthelawoftheevolutionofthepopulation.SteinandUlamstudiedthemovementofpointsinthesimplexundervarioussimple transformationsofthatkindandtheyobtainedverystrangetrajectoriesofthesepoints.Theirexperimentssuggestedtheexistenceofcomplicatedsets,so­called strangeattractors,andothers.Thisbecameaverylivelyfieldofstudy.

Computers,MonteCarloMethod,TheHydrogenBomb PaperswithvonNeumannandRichtmyer(ABA),andwithMetropolis(SNU),proposetheapplicationofcomputersto

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statisticalstudiesbythemethodofrandomsamples.ThisbecameknownastheMonteCarloMethod.Multidimensionalintegralscanoftenbeevaluatedbythis methodevenwhenallothermethodsfail.Inthiswaytheefficiencyofshieldsofnuclearreactors,so­calledneutrontransportproblems,andmanyotherproblemswere solved.BymeansofcomputersUlamalsostudiedtheevolutionofpopulationsofstarsunderthelawofgravitationalattractionandtheevolutionofgenesinthegenetic poolofaspecies,takingintoaccountmutationsandsexualreproduction.Healsoplayedwithsimplerulesleadingtotheinterestingevolutionofsomediscretedynamic systems(ABA). Hisworkonfusionandthehydrogenbombremainsclassified.(SeearticlebyHirschandMathews.) ThelifeandworkofStanislawUlamteachesusthatonecanmakeimportantcontributionstosciencebylettingone'simaginationroamfreelyuponunexploredtopics andbytakingthefullestadvantageoftheuniversalityofthelanguageofmathematics. JANMYCIELSKI, SANTAFE,1990

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PROLOGUE AtdusktheplanefromWashingtontoAlbuquerqueapproachedtheSandiaMountainrangeatthefootofwhichnestlesthecityofAlbuquerque.Sometenminutes beforethelanding,thelightsofthecityofSantaFebecamevisibleinthedistance.OntheWesternhorizonloomedthemysteriousmassofthevolcanicJemez Mountains.ItwasperhapsthehundredthtimeIwasreturningfromWashington,NewYork,orCalifornia,whereLosAlamosaffairsorsomeothergovernmentor academicbusinesstookmealmosteverymonth. MythoughtstraveledbacktomyfirstarrivalinNewMexicoinJanuaryof1944.IwasayoungprofessorattheUniversityofWisconsinandhadbeencalledto participateinaproject,theexactnatureofwhichcouldnotbedivulgedatthetime.AllIwastoldwashowtogettotheLosAlamosarea—atrainstationnamed LamynearSantaFe. Ifsomeonehadprophesiedsomeforty­fiveyearsago

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thatI,ayoung''pure"mathematicianfromLwów,Poland,wouldspendagoodpartofmyadultlifeinNewMexico—astatewhosenameandexistenceIwasnot evenawareofwhenIlivedinEurope—Iwouldhavedismissedtheideaasinconceivable. IfoundmyselfrecollectingmychildhoodinPoland,mystudies,mypreoccupationwithmathematicsevenatanearlyage,andhowmyinterestinphysicsledmeto enlargemyscientificcuriosity,whichinturn—byaseriesofaccidentsandchance—ledtoacalltojointheLosAlamosProject.ThenatureoftheworkthereIonly vaguelyguessedwhenmyfriendJohnvonNeumannaskedmetojoinhimandotherphysicistsatastrangeplace."WestoftheRioGrande,"wasallhecouldtellme whenImethimbetweentrainsatUnionStationinChicago. TheplanelandedatAlbuquerque.Itookmybags,walkedahundredyardsacrossaparkingarea,andclimbedintothesmallplanethatcommutedseveraltimesaday betweenAlbuquerqueandasinglerunwayatanaltitudeof7300feetontheLosAlamosmesa. VonNeumann,oneofthegreatestmathematiciansofthefirsthalfofthetwentiethcentury,wasthepersonwhohadbeenresponsibleformycomingtothiscountryin 1936.Wehadcorrespondedsince1934aboutsomeabstrusequestionsofpuremathematics.ItwasinthisfieldthatIearlymadeanameformyself;vonNeumann, workinginsimilarareas,invitedmetovisitthenewlyestablishedInstituteforAdvancedStudiesinPrinceton—aplacewellknowntothegeneralpublicbecauseoneof itsfirstprofessorswasAlbertEinstein.VonNeumannhimselfwasoneoftheyoungestprofessorsatPrinceton.Hewasalreadyfamousforhisworkinthefoundations ofmathematicsandlogic.Yearslater,hewastobecomeoneofthepioneersinthedevelopmentofelectroniccomputers. AtonetimeIhadundertakentowriteabookonvonNeumann'sscientificlife.Intryingtoplanit,IthoughtofhowI,alongwithmanyothers,hadbeeninfluencedby him;

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andhowthisman,andsomeothersIknew,workinginthepurelyabstractrealmofmathematicsandtheoreticalphysicshadchangedaspectsoftheworldaswenow knowit. Memoriesofmyownworkinscience,ofmystudiesandearlyresearch,oftheendlesshoursspentincafésinmyhometowndiscussingmathematicswithfellow mathematicians,ofmycomingtotheUnitedStates,lecturingatPrincetonandHarvard,becameinterwoveninaninextricablewaywithrecollectionsofvonNeumann's lifeandlaterevents. WhenIstartedtoorganizemythoughts,Irealizedthatuptothattime—itwasabout1966,Ithink—thereexistedfewdescriptionsoftheunusualclimateinwhichthe birthoftheatomicagetookplace.Officialhistoriesdonotgivetherealmotivationsorgointotheinnerfeelings,doubts,convictions,determination,andhopesofthe individualswhoforovertwoyearslivedunderunusualconditions.Asetofflatpictures,theygiveatbestonlytheessentialfacts. ThinkingofallthisinthelittleplanefromAlbuquerquetoLosAlamos,IrememberedhowJulesVerneandH.G.Wellshadinfluencedmeinmychildhoodinthe booksIreadinPolishtranslation.Eveninmyboyishdreams,IdidnotimaginethatsomedayIwouldtakepartinequallyfantasticundertakings. TheresultofallthesereflectionswasthatinsteadofwritingalifeofvonNeumann,Ihaveundertakentodescribemypersonalhistory,aswellaswhatIknowofa numberofotherscientistswhoalsobecameinvolvedinthegreattechnologicalachievementsofthisage. AsIhavealreadymentioned,Ibeganasapuremathematician.InLosAlamosImetphysicistsandother"natural"scientists,andconsortedmainly,ifnotexclusively, withtheoreticians.Itisstillanunendingsourceofsurpriseformetoseehowafewscribblesonablackboardoronasheetofpapercouldchangethecourseofhuman affairs. Ibecameinvolvedintheworkontheatomicbomb,thenintheworkonthehydrogenbomb,butmostofmylifehas

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beenspentinmoretheoreticalrealms.MyfriendOttoFrisch,thediscovererofthepossibilityofchainreactionsfromfission,inanarticleinTheBulletinofthe AtomicScientistsdescribinghisfirstimpressionsofLosAlamosuponarrivingtherefromembattledBritain,wrote: "CertainlyIhaveneverfoundsuchaconcentrationofinterestingpeopleinoneplace.IntheeveningIfeltIcouldwalkintoanyhouseatrandomandwouldfind congenialpeopleengagedinmusicmakingorinstimulatingdebate....IalsometStanUlamearlyon,abrilliantPolishtopologistwithacharmingFrenchwife.Atonce hetoldmethathewasapuremathematicianwhohadsunksolowthathislatestpaperactuallycontainednumberswithdecimalpoints!" Littlehasbeenwrittenaboutthelivesofthepeopleresponsibleforsomuchinscienceandinthebirthofthenuclearageandthespaceage:vonNeumann,Fermi,and numerousothermathematiciansandphysicists.ButhereIwanttorecountalsothemoreabstractandphilosophicallydecisiveinfluenceswhichcamefrommathematics itself.NameslikeStefanBanach,G.D.Birkhoff,andDavidHilbertarevirtuallyunknowntothegeneralpublic,andyetitisthesemen,alongwithEinstein,Fermiand afewothersequallyfamous,whowereindispensabletowhattwentieth­centurysciencehasaccomplished.

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PARTI BECOMINGAMATHEMATICIANINPOLAND

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Chapter1 Childhood 1909–1927 Myfather,JozefUlam,wasalawyer.HewasborninLwów,Poland,in1877.AtthetimeofhisbirththecitywasthecapitaloftheprovinceofGalicia,partofthe Austro­HungarianEmpire.WhenIwasbornin1909thiswasstilltrue. Hisfather,mygrandfather,wasanarchitectandabuildingcontractor.Iunderstandthatmygreat­grandfatherhadcometoLwówfromVenice. Mymother,AnnaAuerbach,wasborninStryj,asmalltownsomesixtymilessouthofLwów,neartheCarpathianMountains.Herfatherwasanindustrialistwho dealtinsteelandrepresentedfactoriesinGaliciaandHungary. OneofmyearliestmemoriesisofsittingonawindowsillwithmyfatherandlookingoutatastreetonwhichtherewasagreatparadehonoringtheCrownPrince, whowasvisitingLwów.Iwasnotquitethreeyearsold. Irememberwhenmysisterwasborn.Iwastoldalittle

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girlhadarrived,andIfelt—itishardtodescribe—somehowgrownup.Iwasthree. WhenIwasfour,Irememberjumpingaroundonanorientalruglookingdownatitsintricatepatterns.Iremembermyfather'stoweringfigurestandingbesideme,and Inoticedthathesmiled.Ifelt,"HesmilesbecausehethinksIamchildish,butIknowthesearecuriouspatterns."Ididnotthinkinthoseverywords,butIampretty certainthatitwasnotathoughtthatcametomelater.Idefinitelyfelt,"Iknowsomethingmyfatherdoesnotknow.PerhapsIknowbetterthanmyfather." IalsohavethememoryofatriptoVenicewiththefamily.Wewereonavaporettoonacanal,andIhadaballoonwhichfelloverboard.Asitbobbedalongtheside oftheboat,myfathertriedtofishitoutwiththecrookedendofhiswalkingstickbutfailed.Iwasconsoledbybeingallowedtoselectasouvenirmodelofagondola madeofVenetianbeadsandstillrememberthefeelingofprideatbeinggivensuchatask. IrememberthebeginningofthefirstWorldWar.Asaboy,IwasaCentralPowerspatriotwhenAustria,Germany,andBulgaria—the"CentralPowers"—were fightingagainstFrance,England,Russia,andItaly.MostofthePolish­speakingpeoplewerenationalisticandanti­Austrian,butnevertheless,atabouttheageofeightI wrotealittlepoemaboutthegreatvictoriesoftheAustrianandGermanarmies. Earlyin1914,theRussiantroopsadvancedintoGaliciaandoccupiedLwów.Myfamilyleft,takingrefugeinVienna.ThereIlearnedGerman,butmynative language—thelanguagewespokeathome—wasPolish. WelivedinahotelacrossfromSt.Stephen'scathedral.ThestrangethingisthateventhoughIvisitedViennamanytimesafterwards,Ididnotactuallyrecognizethis buildingagainuntilonedayin1966whileIwaswalkingthroughthestreetswithmywife.PerhapsbecauseweweretalkingaboutmychildhoodIsuddenly remembereditandpointed

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itouttoher.Withthisanumberofothermemoriesburiedforoverfiftyyearssurfaced. Onthesamevisit,whilewalkingthroughthePratergardens,thesightofanoutdoorcafésuddenlybroughtbackthememoryofhowIhadoncechokedinthewind withasortofasthmaticreactioninfrontofthatverycafé—afeelingthatIwasnottoexperienceagainuntilmanyyearslaterinMadison,Wisconsin.Curiouslythe subsequentsensationdidnotmakemerecallthechildhoodepisode.ItisonlywhenIwasatthatveryspotmanyyearslaterthatthissensorymemoryreturnedasa resultofthevisualassociation. IwillnottrytodescribethemoodofViennaasseenthroughtheeyesofasix­year­old.Iworeasortofmilitarycap;whenanofficersalutedmeonKärntnerStrasse (oneofthemainstreetsofVienna)IremembervividlythatIwasabsolutelydelighted.ButwhensomebodymentionedthattheUnitedStateswouldhavetenthousand airplanes(therewassucharumor)IbegantohavedoubtsaboutthevictoryoftheCentralPowers. AtaboutthistimeinViennaIlearnedtoread.Likesomuchoflearningthroughoutmylife,atfirstitwasanunpleasant—adifficult,somewhatpainfulexperience.After awhile,everythingfellintoplaceandbecameeasy.Irememberwalkingthestreetsreadingallthesignsaloudwithgreatpleasure,probablyannoyingmyparents. MyfatherwasanofficerintheAustrianArmyattachedtomilitaryheadquarters,andwetraveledfrequently.ForawhilewelivedinMärischOstrau,andIwentto schoolthereforatime.Inschoolwehadtolearnthemultiplicationtables,andIfoundlearningarithmeticmildlypainful.OnceIwaskepthomewithacoldjustaswe wereatsixtimesseven.IwassurethattherestoftheclasswouldbeattwelvetimesfifteenbythetimeIwentback.IthinkIwenttotentimestenbymyself.Therest ofthetimeIhadtutors,forwetraveledsomuchitwasnotpossibletoattendschoolregularly.

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Ialsorememberhowmyfatherwouldsometimereadtomefromachildren'seditionofCervantes'DonQuixote.Episodesthatnowseemonlymildlyfunnytome,I consideredhilarious.IthoughtthedescriptionofDonQuixote'sfightwiththewindmillsthefunniestthingimaginable. Thesearevisualpictures,notnostalgicreallybutbearingadefinitetaste,andtheyleaveadefiniteflavorofassociationsinthememory.Theycarrywiththema consciousnessofdifferentintensities,differentcolors,differentcompositions,mixedwithfeelingswhicharenotexplicit—ofwell­beingorofdoubt.Theycertainlyplay simultaneouslyonmanyphysicallyseparatepartsinthebrainandproduceafeelingperhapsakintoamelody.ItisareconstructionofhowIfelt.Peopleoftenretain theserandompictures,andthestrangethingisthattheypersistthroughoutone'slife. Certainscenesareeasierofaccess,butthereareprobablymanyotherimpressionswhichcontinuetoexist:Experimentshavere­createdcertainscenesfromthepast whenareasofapatient'sbrainweretouchedwithaneedleduringanoperation.Thescenesthatcanbesummonedupfromone'smemoryatwillhaveacolororflavor whichdoesnotseemtochangewithtime.Theirre­creationbyrecollectiondoesnotseemtochangethemorrefreshthem.AsfarasIcantellwhenItrytoobservein myselfthechainofsyllogismsinitiatedbytheseimpressions,theyarequiteanalogousnowastowhattheywerewhenIwaslittle.IfIlooknowatanobject,likea chair,oratree,oratelegraphwire,itinitiatesatrainofthought.AnditseemstomethatthesuccessionoflinkedmemoriesarequitethesameasthoseIremember whenIwasfiveorsix.WhenIlookatatelegraphwire,Irememberverywellitgavemeasortofabstractormathematicalimpulse.Iwonderedwhatelsecoulddo that.Itwasanattemptatgeneralization. Perhapsthestoreofmemoryinthehumanbrainistoalargeextentalreadyformedataveryearlyage,andexternalstimuliinitiateaprocessofrecordingand classifyingtheim­

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pressionsalongchannelswhichexistinlargenumbersinveryearlychildhood. Tolearnhowthingsarefiledinthememory,itobviouslyhelpstoanalyzeone'sthoughts.Tounderstandhowoneunderstandsatext,oranewmethod,ora mathematicalproof,itisinterestingtotrytoconsciouslyperceivethetemporalorderandtheinnerlogic.Professionalsoreveninterestedamateurshavenotdone enoughinthisareatojudgebywhatIhavereadonthenatureofmemory.Itseemstomethatmorecouldbedonetoeliciteveninpartthenatureofassociations,with computersprovidingthemeansforexperimentation.Suchastudywouldhavetoinvolveagradationofnotions,ofsymbols,ofclassesofsymbols,ofclassesof classes,andsoon,inthesamewaythatthecomplexityofmathematicalorphysicalstructuresisinvestigated. Theremustbeatricktothetrainofthought,arecursiveformula.Agroupofneuronsstartsworkingautomatically,sometimeswithoutexternalimpulse.Itisakindof iterativeprocesswithagrowingpattern.Itwandersaboutinthebrain,andthewayithappensmustdependonthememoryofsimilarpatterns. Verylittleisknownaboutthis.Perhapsbeforeahundredyearshavepassedthiswillallbepartofafascinatingnewscience.Itwasnotsolongagothatscientistslike JohnvonNeumannbegantoexamineanalogiesbetweentheoperationofthebrainandthatofthecomputer.Earlier,peoplehadthoughttheheartwastheseatof thought;thentheroleofthebrainbecamemoreevident.Perhapsitactuallydependsonallthesenses. Weareaccustomedtothinkofthinkingasalinearexperience,aswhenwesay"train"ofthought.Butsubconsciousthinkingmaybemuchmorecomplicated.Justas onehassimultaneousvisualimpressionsontheretina,mighttherenotbesimultaneous,parallel,independentlyoriginated,abstractimpressionsinthebrainitself? Somethinggoesoninourheadsinprocesseswhicharenotsimplystrungouton

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oneline.Inthefuture,theremightbeatheoryofamemorysearch,notbyonesensorgoingaround,butperhapsmorelikeseveralsearcherslookingforsomeonelost inaforest.Itisaproblemofpursuitandofsearch—oneofthegreatestareasofcombinatorics. Whathappenswhenonesuddenlyremembersaforgottenwordorname?Whatdoesonedowhenonetriestorememberit?Subconsciouslysomethingisturning. Morethanonerouteisfollowed:onetriesbysoundorletters,longwordsorshortwords.Thatmustmeanthatthewordisfiledinmultiplestorage.Ifitwereonlyin oneplacetherewouldbenowaytorecoverit.Timeisaparameter,too,andalthoughintheconsciousthereseemstobeonlyonetime,theremaybemanyinthe subconscious.Thenthereisthemechanismofsynthesizerorsummarizer.Couldoneintroduceanautomaticsearchsystem,aningenioussystemwhichdoesnotgo througheverythingbutscanstherelevantelements? ButIhavedigressedenoughintheseobservationsonmemory.Letmenowreturntothisaccountofmylife.IonlywishthatIcouldhavesomeofVladimirNabokov's abilitytoevokepanoramasofmemoriesfromafewpicturesofthepast.Indeedonecansaythatanartistdepictstheessentialfunctionsorpropertiesofawholesetof impressionsontheretina.Itisthesethatthebrainsummarizesandstoresinthememory,justasacaricaturistcanconveytheessentialsofafacewithjustafew strokes.Mathematicallyspeaking,thesearetheglobalcharacteristicsofthefunctionorthefigureofasetofpoints.InthismoreprosaicaccountIwilldescribemerely themoreformalpoints. In1918wereturnedtoLwów,whichhadbecomepartofthenewlyformedRepublicofPoland.InNovemberofthatyearthe,Ukrainiansbesiegedthecity,which wasdefendedbyasmallnumberofPolishsoldiersandarmedcivilians.Ourhousewasinarelativelysafepartoftown,eventhoughoccasionalartilleryshellsstruck nearby.Becauseourhousewassafer,manyofourrelativescametostaywithus.There

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musthavebeensomethirtyofthem,halfbeingchildren.Therewerenotnearlyenoughbeds,ofcourse,andIrememberpeoplesleepingeverywhereonrolledrugson thefloor.Duringtheshellingwehadtogotothebasement.Istillrememberinsistingontyingmyshoeswhilemymotherwaspressingmetohurrydownstairs.Forthe adultsitmusthavebeenastrenuoustimetosaytheleast,butnotforus.Strangelyenough,mymemoriesofthesedaysareofthefunIhadplaying,hiding,learning cardgameswiththechildrenforthetwoweeksbeforethesiegewasliftedwiththearrivalofanotherPolisharmyfromFrance.Thisbroketheringofbesiegers.For childrenwartimememoriesarenotalwaystraumatic. DuringthePolish­Russianwarin1920thecitywasthreatenedagain.Budenny'scavalrypenetratedtowithinfiftymiles,butPilsudski'svictoryontheWarsawfront savedthesouthernfrontandthewarended. Attheageoftenin1919Ipassedtheentranceexaminationtothegymnasium.ThiswasasecondaryschoolpatternedaftertheGermangymnasiaandtheFrench lycées.Instructionusuallytookeightyears.IwasanAstudent,exceptinpenmanshipanddrawing,butdidnotstudymuch. Oneofthegapsinmyeducationwasinchemistry.Wedidnothavemuchofitinschoolandfiftyyearslater,nowthatIaminterestedinbiology,thishandicapsmein mystudiesofelementarybiochemistry. AboutthistimeIalsodiscoveredthatIdidnothavequitenormalbinocularvision.Ithappenedinthefollowingway:theboysintheclasshadbeenlinedupforaneye examination.Awaitingmyturntoreadthecharts,Icoveredmyeyeswithmyhand.InoticedwithhorrorthatIcouldonlyreadthelargestletterswithmyrighteye. ThismademeafraidthatIwouldbekeptafterschool,soImemorizedtheletters.IthinkitwasthefirsttimeinmylifewhenIconsciouslycheated.Whenmyturn cameI''read"satisfactorilyandwasletoff,butIknewmyeyesweredifferent,onewasmyopic.Theother,normal,laterbecamepresbyotic.This

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condition,ratherrarebutwellknown,isapparentlyhereditary.Istillhaveneverwornglasses,althoughIhavetobendclosetotheprintedtexttoreadwithmymyopic eye.IamnotnormallyawarewhicheyeIuse;oncelaterinlifeadoctorinMadisontoldmethatthisconditionissometimesbetterthannormal,foroneortheother eyeisrestingwhiletheotherisinuse.Iwonderifmypeculiareyesight,inadditiontoaffectingmyreadinghabits,mayalsohaveaffectedmyhabitsofthought. WhenItrytorememberhowIstartedtodevelopmyinterestinscienceIhavetogobacktocertainpicturesinapopularbookonastronomyIhad.Itwasatextbook calledAstronomyofFixedStars,byMartinErnst,aprofessorofastronomyattheUniversityofLwów.InitwasareproductionofaportraitofSirIsaacNewton.I wasnineortenatthetime,andatthatageachilddoesnotreactconsciouslytothebeautyofaface.YetIrememberdistinctlythatIconsideredthisportrait— especiallytheeyes—assomethingmarvelous.Amixtureofphysicalattractionandafeelingofthemysteriousemanatedfromhisface.LaterIlearneditwasthe GeoffreyKnellerportraitofNewtonasayoungman,withhairtohisshouldersandanopenshirt.OtherillustrationsIdistinctlyrememberwereoftheringsofSaturn andofthebeltsofJupiter.Thesegavemeacertainfeelingofwonder,theflavorofwhichishardtodescribesinceitissometimesassociatedwithnonvisual impressionssuchasthefeelingonegetsfromanexquisiteexampleofscientificreasoning.Butitreappears,fromtimetotime,eveninolderage,justasafamiliarscent willreappear.Occasionallyanodorwillcomeback,bringingcoincidentmemoriesofchildhoodoryouth. Readingdescriptionsofastronomicalphenomenatodaybringsbacktomethesevisualmemories,andtheyreappearwithanostalgic(notmelancholybutrather pleasant)feeling,whennewthoughtscomeaboutoranewdesireformentalworksuddenlyemerges. ThehighpointofmyinterestinastronomyandanunforgettableemotionalexperiencecamewhenmyuncleSzy­

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monUlamgavemealittletelescope.Itwasoneofthecopper­orbronze­tubevarietyand,Ibelieve,arefractorwithatwo­inchobjective. Tothisday,wheneverIseeaninstrumentofthiskindinantiqueshops,nostalgiaovercomesme,andafterallthesedecadesmythoughtsstillturntovisionsofthe celestialwondersandnewastronomicalproblems. Atthattime,Iwasintriguedbythingswhichwerenotwellunderstood—forexample,thequestionoftheshorteningoftheperiodofEncke'scomet.Itwasknownthat thiscometirregularlyandmysteriouslyshortensitsthree­yearperiodofmotionaroundthesun.Nineteenth­centuryastronomersmadeseveralattemptstoaccountfor thisasbeingcausedbyfrictionorbythepresenceofsomenewinvisiblebodyinspace.Itexcitedmethatnobodyreallyknewtheanswer.Ispeculatedwhetherthe 1/r2lawofattractionofNewtonwasnotquiteexact.Itriedtoimaginehowitcouldaffecttheperiodofthecometiftheexponentwasslightlydifferentfrom2, imaginingwhattheresultwouldbeatvariousdistances.Itwasanattempttocalculate,notbynumbersandsymbols,butbyalmosttactilefeelingscombinedwith reasoning,averycuriousmentaleffort. Nostarcouldbelargeenoughforme.BetelgeuseandAntareswerebelievedtobemuchlargerthanthesun(eventhoughatthetimenoprecisedatawereavailable) andtheirdistancesweregiven,aswereparallaxesofmanystars.IhadmemorizedthenamesofconstellationsandtheindividualArabicnamesofstarsandtheir distancesandluminosities.Ialsoknewthedoublestars. InadditiontotheexcitingErnstbookanother,entitledPlanetsandtheConditionsofLifeonThem,wasstrange.SoonIhadsomeeightortenastronomybooksin mylibrary,includingthemarvelousNewcomb­EngelmannAstronomieinGerman.TheBode­Titiusformulaor"law"ofplanetarydistancesalsofascinatedme, inspiringmetobecomeanastronomerorphysicist.Thiswasaboutthetimewhen,attheageofelevenorso,Iinscribedmynameinanotebook,"S.

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Ulam,astronomer,physicistandmathematician."Myloveforastronomyhasneverceased;Ibelieveitisoneoftheavenuesthatbroughtmetomathematics. Fromtoday'sperspectiveLwówmayseemtohavebeenaprovincialcity,butthisisnotso.Frequentlecturesbyscientistswereheldforthegeneralpublic,inwhich suchtopicsasnewdiscoveriesinastronomy,thenewphysicsandthetheoryofrelativitywerecovered.Theseappealedtolawyers,doctors,businessmen,andother laymen. OtherpopularlecturetopicswereFreudandpsychoanalysis.Relativitytheorywas,ofcourse,muchmoredifficult. Around1919–1920somuchwaswritteninnewspapersandmagazinesaboutthetheoryofrelativitythatIdecidedtofindoutwhatitwasallabout.Iwenttosomeof thepopulartalksonrelativity.Ididnotreallyunderstandanyofthedetails,butIhadagoodideaofthemainthrustofthetheory.Almostlikelearningalanguagein childhood,onedevelopstheabilitytospeakitwithoutknowinganythingaboutgrammar.Curiouslyenough,itispossibleevenintheexactsciencestohaveanideaof thegistofsomethingwithouthavingacompleteunderstandingofthebasics.Iunderstoodtheschemaofspecialrelativityandevensomeofitsconsequenceswithout beingabletoverifythedetailsmathematically.Ibelievethatso­calledunderstandingisnotayes­or­noproposition.Butwedon'tyethavethetechniqueofdefining theselevelsorthedepthoftheknowledgeofreasons. Thisinterestbecameknownamongfriendsofmyfather,whoremarkedthatI"understood"thetheoryofrelativity.Myfatherwouldsay,"Thelittleboyseemsto understandEinstein!"ThisgavemeareputationIfeltIhadtomaintain,eventhoughIknewthatIdidnotgenuinelyunderstandanyofthedetails.Nevertheless,this wasthebeginningofmyreputationasa"brightchild."Thisencouragedmetofurtherstudyofpopularsciencebooks—anexperience

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Iamsureiscommontomanychildrenwholatergrowuptobescientists. Howachildacquiresthehabitsandinterestswhichplaysuchadecisiveroleindetermininghisfuturehasnotbeensufficientlyinvestigated."Plagiarism"—the mysteriousabilityofachildtoimitateorcopyexternalimpressionssuchasthemother'ssmile—isonepossibleexplanation.Anotherisinborncuriosity:whydoesone seeknewexperiencesinsteadofmerelyreactingtostimuli? Inclinationsmaybepartoftheinheritedsystemofconnectionsinthebrain,agenetictraitthatmaynotevendependonthephysicalarrangementofneurons. Apparentlyheadachesarerelatedtotheeasewithwhichbloodcirculatesinthebrain,whichdependsonwhetherthebloodvesselsarewideornarrow.Perhapsitis the"plumbing"thatisimportant,ratherthanthearrangementoftheneuronsnormallyassociatedwiththeseatofthinking. Anotherdeterminingfactormaybeinitialaccidentsofsuccessorfailureinanewpursuit.Ibelievethatthequalityofmemorydevelopssimilarlyasaresultofinitial accidents,randomexternalinfluences,oraluckycombinationofthetwo. Considerthetalentforchess,forexample.JoséCapablancalearnedthegameattheageofsixbywatchinghisfatheranduncleplay.Hedevelopedtheabilitytoplay naturally,effortlessly,thewayachildlearnstospeakascomparedwiththestrugglesadultshaveinlearningnewsubjects.Otherfamouschessplayersalsofirst becameinterestedbywatchingtheirrelativesplay.Whentheytried,perhapsachanceinitialsuccessencouragedthemtopursue.Nothingsucceedslikesuccess,itis wellknown,especiallyinearlyyouth. Ilearnedchessfrommyfather.Hehadalittlepaper­boundbookonthesubjectandusedtotellmeaboutsomeofthefamousgamesitdescribed.Themovesofthe knightfascinatedme,especiallythewaytwoenemypiecescanbe

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threatenedsimultaneouslywithoneknight.Althoughitisasimplestratagem,Ithoughtitwasmarvelous,andIhavelovedthegameeversince. Couldthesameprocessapplytothetalentformathematics?Achildbychancehassomesatisfyingexperienceswithnumbers;thenheexperimentsfurtherand enlargeshismemorybybuildinguphisstoreofexperiences. Ihadmathematicalcuriosityveryearly.MyfatherhadinhislibraryawonderfulseriesofGermanpaperbackbooks—Reklam,theywerecalled.OnewasEuler's Algebra.IlookedatitwhenIwasperhapstenoreleven,anditgavemeamysteriousfeeling.Thesymbolslookedlikemagicsigns;IwonderedwhetheronedayI couldunderstandthem.Thisprobablycontributedtothedevelopmentofmymathematicalcuriosity.Idiscoveredbymyselfhowtosolvequadraticequations.I rememberthatIdidthisbyanincredibleconcentrationandalmostpainfulandnot­quite­consciouseffort.WhatIdidamountedtocompletingthesquareinmyhead withoutpaperorpencil. Inhighschool,Iwasstimulatedbythenotionoftheproblemoftheexistenceofoddperfectnumbers.Anintegerisperfectifitisequaltothesumofallitsdivisors includingonebutnotitself.Forinstance:6=1+2+3isperfect.Sois28=1+2+4+7+14.Youmayask:doesthereexistaperfectnumberthatisodd?The answerisunknowntothisday. Ingeneral,themathematicsclassesdidnotsatisfyme.Theyweredry,andIdidnotliketohavetomemorizecertainformalprocedures.Ipreferredreadingonmy own. AtaboutfifteenIcameuponatreatiseontheinfinitesimalcalculusinabookbyGerhardtKowalevski.Ididnothaveenoughpreparationinanalyticgeometryoreven intrigonometry,buttheideaoflimits,thedefinitionsofrealnumbers,thenotionofderivativesandintegrationpuzzledandexcitedmegreatly.Idecidedtoreadapage ortwoadayandattempttolearnthenecessaryfactsabouttrigonometryandanalyticgeometryfromotherbooks. Ifoundtwootherbooksinasecondhandbookstore.

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Theseintriguedandfascinatedmemorethananythingelseformanyyearstocome:Sierpinski'sTheoryofSetsandamonographonnumbertheory.Attheageof seventeenIknewasmuchormoreelementarynumbertheorythanIdonow. IalsoreadabookbythemathematicianHugoSteinhausentitledWhatIsandWhatIsNotMathematicsandinPolishtranslationPoincaré'swonderfulLaScience etl'Hypothèse,LaScienceetlaMéthode,LaValeurdelaScience,andhisDernièresPensées.Theirliteraryquality,nottomentionthescience,wasadmirable. Poincarémoldedportionsofmyscientificthinking.Readingoneof'hisbookstodaydemonstrateshowmanywonderfultruthshaveremained,althougheverythingin mathematicshaschangedalmostbeyondrecognitionandinphysicsperhapsevenmoreso.IadmiredSteinhaus'sbookalmostasmuch,foritgavemanyexamplesof actualmathematicalproblems. Themathematicstaughtinschoolwaslimitedtoalgebra,trigonometry,andtheverybeginningofanalyticgeometry.Intheseventhandeighthclasses,wherethe studentsweresixteenandseventeen,therewasacourseonelementarylogicandasurveyofhistoryofphilosophy.Theteacher,ProfessorZawirski,wasareal scholar,alecturerattheUniversityandaverystimulatingman.Hegaveusglimpsesofrecentdevelopmentsinadvancedmodernlogic.HavingstudiedSierpinski's booksontheside,Iwasabletoengagehimindiscussionsofsettheoryduringrecessandinhisoffice.Iwasworkingonsomeproblemsontransfinitenumbersandon theproblemofthecontinuumhypothesis. Ialsoengagedinwildmathematicaldiscussions,formulatingvastandnewprojects,newproblems,theoriesandmethodsborderingonthefantastic,withaboynamed Metzger,somethreeorfouryearsmysenior.Hehadbeendirectedtowardmebyfriendsof'myfatherwhoknewthathetoohadagreatinterestinmathematics. Metzgerwasshort,rotund,blondish,atypicalliberatedghettoJew.LaterIsawayouthfulportraitofHeinewhichremindedmeofhis

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face.Peopleofhistypecanstillbefoundoccasionally.Theyexhibitamateurism,evenabouttheveryfoundationsofarithmetic.Wediscussed"aniterativecalculus"on thebasisofpracticallynoknowledgeoftheexistingmathematicalmaterial.Hewas"crazy"andfulloftheurgetoinnovatewhichissoJewish.StefanBanachonce pointedoutthatitischaracteristicofcertainJewsalwaystotrytochangetheestablishedschemeofthings—Jesus,Marx,Freud,Cantor.Onaverysmallscale Metzgershowedthistendency.Hadhehadabettereducationhemighthavedonegoodthings.HeobviouslycamefromaverypoorfamilyandhisPolishhada strong,gutturalaccent.Afterafewmonthsheabruptlyvanishedfrommyken.ThisisthefirsttimeIhavethoughtabouthiminalltheseyears.Perhapsheisalive.This memoryofMetzgerandourdiscussionsbringsbacktheverysmellandcolorofthe"abstractions"weexchanged. Strangelyenough,atthisyouthfulandimmatureageIwasalsooccasionallytryingtoanalyzemyownthinkingprocesses.Itriedtomakemyselfmoreawareofthem byperiodicallygoingbackeveryfewsecondstoseewhatitwasthatmoldedthetrainofthought.Needlesstosay,Iwasfullyawareofthefactthatthereisadangerin indulgingtoomuchandtoofrequentlyinsuchintrospection. Sofar,theimageIhadformedofastronomersandscientists,andofmathematiciansinparticular,camealmostexclusivelyfrommyreading.Igotmyfirst"live" impressionswhenIwenttoaseriesofpopularmathematicslecturesin1926.OnsuccessivedaysthereweretalksbyHugoSteinhaus,StanislawRuziewicz,Stefan Banach,andperhapsothers.Myfirstsurprisewastodiscoverhowyoungtheywere.HavingheardandreadoftheirachievementsIreallyexpectedbeardedold scholars.Ilistenedavidlytotheirtalks.YoungasIwas,myimpressionofBanachwasthatherewasahomespungenius.Thisfirstimpression—deepened,enriched, andtransformed,ofcourse—remainedduringmysubsequentlongacquaintance,collaboration,andfriendshipwithhim.

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Thenin1927,ZawirskitoldmeacongressofmathematicianswastotakeplaceinLwówandforeignscholarshadbeeninvited.Headdedthatayouthfuland extremelybrilliantmathematiciannamedJohnvonNeumannwastogivealecture.ThiswasthefirsttimeIheardthename.Unfortunately,Icouldnotattendthese lecturesforIwasinthemidstofmyownmatriculationexaminationsattheGymnasium. Still,myinterestsinsciencedidnottakeallofmytime.IavidlyreadPolishliterature,aswellaswritersasdiverseasTolstoy,JulesVerne,KarlMay,H.G.Wells,and AnatoleFrance.AsaboyIpreferredbiographiesandadventurestories. Besidesthesemorecerebralactivities,Iengagedactivelyinsports.BeginningataboutfourteenIplayedvariouspositionsinsoccerwithmyclassmates:goalie,right forwardandothers.Istartedplayingtennis,too,andwasactiveintrackandfield. AfterschoolIplayedcardswithmyclassmates.Weplayedbridgeandasimplevarietyofpokerforsmallstakes.Inpokertheolderboyswonmostofthetime.One oftheabilitiesthatapparentlydoesnotdecreasebutratherimproveswithageisaprimitivetypeofelementaryshrewdness.Iplayedchessalso,twoorthreetimesa week.AlthoughIdon'tthinkIeverhadtoomuchtalentforthegame,Icertainlyhadamorethanaveragefeelingforpositions,andIprobablywasoneofthebest playersinmygroup.Likemathematics,chessisoneofthethingswhereconstantpractice,constantthinking,andimagining,andstudyingarenecessarytoachievea masteryofthegame. In1927Ipassedmythreedaymatriculationexaminationsandaperiodofindecisionbegan.Thechoiceofafuturecareerwasnoteasy.Myfather,whohadwanted metobecomealawyersoIcouldtakeoverhislargepractice,nowrecognizedthatmyinclinationslayinotherdirections.Besides,therewasnoshortageoflawyersin Lwów.Thethoughtofauniversitycareerwasattractive,butprofessorial

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positionswererareandhardtoobtain,especiallyforpeoplewithJewishbackgroundslikemyself.Consequently,Ilookedforacourseofstudieswhichwouldleadto somethingpracticalandatthesametimewouldbeconnectedwithscience.Myparentsurgedmetobecomeanengineer,andsoIappliedforadmissionattheLwów PolytechnicInstituteasastudentofeithermechanicalorelectricalengineering.

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Chapter2 StudentYears 1927–1933 Inthefallof1927IbeganattendinglecturesatthePolytechnicInstituteintheDepartmentofGeneralStudies,becausethequotaofElectricalEngineeringalreadywas full.Theleveloftheinstructionwasobviouslyhigherthanthatathighschool,buthavingreadPoincaréandsomespecialmathematicaltreatises,Inaivelyexpected everylecturetobeamasterpieceofstyleandexposition.Ofcourse,Iwasdisappointed. AsIknewmanyofthesubjectsinmathematicsfrommystudies,Ibegantoattendasecond­yearcourseasanauditor.Itwasinsettheoryandgivenbyayoung professorfreshfromWarsaw,KazimirKuratowski,astudentofSierpinski,Mazurkiewicz,andJaniszewski.Hewasafreshmanprofessor,sotospeak,andIa freshmanstudent.FromtheveryfirstlectureIwasenchantedbytheclarity,logic,andpolishofhisexpositionandthematerialhepresented.FromthebeginningI participatedmoreactivelythanmostofthe

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olderstudentsindiscussionswithKuratowski,sinceIknewsomethingofthesubjectfromhavingreadSierpinski'sbook.IthinkhequicklynoticedthatIwasoneof thebetterstudents;afterclasshewouldgivemeindividualattention.ThisishowIstartedonmycareerasamathematician,stimulatedbyKuratowski. SoonIcouldanswersomeofthemoredifficultquestionsinthesettheorycourse,andIbegantoposeotherproblems.RightfromthestartIappreciatedKuratowski's patienceandgenerosityinspendingsomuchtimewithanovice.SeveraltimesaweekIwouldaccompanyhimtohisapartmentatlunchtime,awalkofabouttwenty minutes,duringwhichIaskedinnumerablemathematicalquestions.Yearslater,Kuratowskitoldmethatthequestionsweresometimessignificant,oftenoriginal,and interestingtohim. Mycoursesincludedmathematicalanalysis,calculus,classicalmechanics,descriptivegeometry,andphysics.Betweenclasses,Iwouldsitintheofficesofsomeofthe mathematicsinstructors.AtthattimeIwasperhapsmoreeagerthanatanyothertimeinmylifetodomathematicstotheexclusionofalmostanyotheractivity. ItwastherethatIfirstmetStanislawMazur,whowasayoungassistantattheUniversity.HecametothePolytechnicInstitutetoworkwithOrlics,Nikliborcand Kaczmarz,whowereafewyearshissenior. InconversationswithMazurIbegantolearnaboutproblemsinanalysis.Irememberlonghoursofsittingatadeskandthinkingaboutthequestionswhichhe broachedtomeanddiscussedwiththeothermathematicians.Mazurintroducedmetoadvancedideasofrealvariablefunctiontheoryandthenewfunctionalanalysis. WediscussedsomeofthemorerecentproblemsofBanach,whohaddevelopedanewapproachtothistheory. Banachhimselfwouldappearoccasionally,eventhoughhismainworkwasattheUniversity.Imethimduringthisfirstyear,butouracquaintancebeganinamore meaningful,intimate,andintellectualsenseayearortwolater.

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Severalothermathematicianscouldfrequentlybeseenintheseoffices.Stozek,cheerful,rotund,short,andcompletelybald,wasChairmanoftheDepartmentof GeneralStudies.Thewordstozekmeans''acone"inPolish;helookedmorelikeasphere.Alwaysingoodhumorandjokingincessantly,helovedtoconsume frankfurtersliberallysmearedwithhorseradish,adishwhichhemaintainedcuredmelancholy.(StozekwasoneoftheprofessorsmurderedbytheGermansin1941.) AntoniLomnicki,amathematicianofaristocraticfeatureswhospecializedinprobabilitytheoryanditsapplicationstocartography,hadofficehoursintheserooms. (HetoowasmurderedbytheGermansinLwówin1941.)Hisnephew,ZbigniewLomnicki,laterbecamemygoodfriendandmathematicalcollaborator. Kaczmarz,tallandthin(wholaterwaskilledinmilitaryservicein1940),andNikliborc,shortandrotund,managedtheexercisesectionsofthelargecalculusand differentialequationscourses.TheywereoftenseentogetherandremindedmeofPatandPatachon,twocontemporarycomicfilmactors. IdidnotfeelIwasaregularstudentinthesensethatonemayhavetostudysubjectsoneisnotespeciallyinterestedin.Ontheotherhand,afteralltheseyears,Istill donotfeelmuchlikeanaccomplishedprofessionalmathematician.Iliketotrynewapproachesand,beinganoptimistbynature,hopetheywillsucceed.Ithasnever occurredtometoquestionwhetheramentaleffortwillbewastedorwhetherto"husband"mymentalcapital. Atthebeginningofthesecondsemesterofmyfreshmanyear,Kuratowskitoldmeaboutaprobleminsettheorythatinvolvedtransformationsofsets.Itwas connectedwithawell­knowntheoremofBernstein:if2A=2B,thenA=B,inthearithmeticsenseofinfinitecardinals.ThiswasthefirstproblemonwhichIreally spentarduoushoursofthinking.Ithoughtaboutitinawaywhichnowseemsmysterioustome,notconsciouslyorexplicitlyknowingwhatIwasaim­

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ingat.SoimmersedinsomeaspectswasI,thatIdidnothaveaconsciousoverallview.Nevertheless,Imanagedtoshowbymeansofaconstructionhowtosolve theproblem,devisingamethodofrepresentingbygraphsthedecompositionofsetsandthecorrespondingtransformations.Unbelievably,atthetimeIthoughtIhad inventedtheveryideaofgraphs. IwrotemyfirstpaperonthisinEnglish,whichIknewbetterthanGermanorFrench.Kuratowskicheckeditandtheshortpaperappearedin1928inFundamenta Mathematicae,theleadingPolishmathematicaljournalwhichheedited.Thisgavemeself­confidence. IstillwasnotcertainwhatcareerorcourseofworkIshouldpursue.ThepracticalchancesofbecomingaprofessorofmathematicsinPolandwerealmostnil—there werefewvacanciesattheUniversity.Myfamilywantedmetolearnaprofession,andsoIintendedtotransfertotheDepartmentofElectricalEngineeringformy secondyear.Inthisfieldthechanceofmakingalivingseemedmuchbetter. BeforetheendoftheyearKuratowskimentionedinalectureanotherprobleminsettheory.Itwasontheexistenceofsetfunctionswhichare"subtractive"butnot completelycountablyadditive.Irememberponderingthequestionforweeks.IcanstillfeelthestrainofthinkingandthenumberofattemptsIhadtomake.Igave myselfanultimatum.IfIcouldsolvethisproblem,Iwouldcontinueasamathematician.Ifnot,Iwouldchangetoelectricalengineering. AfterafewweeksIfoundawaytoachieveasolution.IranexcitedlytoKuratowskiandtoldhimaboutmysolution,whichinvolvedtransfiniteinduction.Transfinite inductionhadbeenusedbymathematicalworkersmanytimesinotherconnections;however,IbelievethatthewayinwhichIuseditwasnovel. IthinkKuratowskitookpleasureinmysuccess,encouragingmetocontinueinmathematics.BeforetheendofmyfirstcollegeyearIhadwrittenmysecondpaper, which

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KuratowskipresentedtoFundamenta.Now,thediewascast.Ibegantoconcentrateonthe"impractical"possibilitiesofanacademiccareer.Mostofwhatpeople calldecisionmakingoccursfordefinitereasons.However,Ifeelthatformostofuswhatisultimatelycalleda"decision"isasortofvotetakeninthesubconscious,in whichthemajorityofthereasonsfavoringthedecisionwinout. Duringthesummerof1928whenItookatriptotheBalticcoastofPoland,KuratowskiinvitedmetovisithimonthewayathissummerplacenearWarsaw.Itwas anelegantvillawithatenniscourt.Kuratowskiwasquitegoodattennisinthosedays,andthissurprisedmesincehisfigurewasanythingbutathletic. Onthesix­hourtrainridefromLwówtoWarsawIthoughtalmostwithoutinterruptionaboutproblemsinsettheorywiththeideaofpresentingsomethingthatwould interesthim.Iwasthinkingofwaystodisprovethecontinuumhypothesis,afamousunsolvedprobleminfoundationsofsettheoryandmathematicsformulatedby GeorgCantor,thecreatorofsettheory.Mypresentationwasvague,andKuratowskisoondetectedthis.Nevertheless,wediscusseditsramifications,andsoIwent ontoZoppotwithmyself­confidenceintact. AlfredTarski,nowacelebratedlogicianandprofessoratBerkeley,wasafriendofKuratowskifromWarsaw,whooccasionallyvisitedLwów.Hewasalready knowninternationallyasalogician,buthisworkinthefoundationsofmathematicallogicandsettheorywasalsoimportant.Hehadbeenacandidateforachairof philosophythatwasvacantattheUniversityofLwów.Thechairwentinsteadtoanotherlogician,LeonChwistek,anaccomplishedpainterandauthorofphilosophical treatises,abrother­in­lawofSteinhaus,andwellknownformanyeccentricities.(HediedinMoscowduringthewar.)YearslaterinCambridge,Ihappenedto mentionChwistektoAlfredNorthWhitehead.InthecourseoftheconversationIsaid,"Verystrange,hewasapaintertoo!"WhereuponWhiteheadlaughedoutloud,

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clappedhishandsandexclaimed:"HowBritishofyoutosaythatbeingapainterisstrange."Mrs.Whiteheadjoinedinthelaughter.Averygoodbiographyof ChwistekbyEstriecherhasrecentlyappearedinPoland.ItisafascinatingaccountoftheintellectualandartisticlifeofCracowandLwówfrom1910to1946. OneofmyearlycontactswithTarskiwasaresultofmysecondpaper.InitIhadprovedatheoremonidealsofsetsinsettheory.(MarshallStonelaterproved anotherversionofthissametheorem.)MynoteinFundamentaalsoshowedthepossibilityofdefiningafinitelyadditivemeasurewithtwovalues,0to1,and establishedamaximumprimeidealforsubsetsintheinfiniteset.Inaverylongpaperwhichappearedayearlater,Tarskigotthesameresult.AfterKuratowski pointedouttohimthatitfollowedfrommytheorem,Tarskiacknowledgedthisinafootnote.Inviewofmyyouth,thisseemedtomealittlevictory—an acknowledgmentofmymathematicalpresence. Therewasafeelingamongsomemathematiciansthatlogicisnot"real"mathematics,butmerelyapreparatoryandsomewhatalienart.Today,thisfeelingis disappearingasaresultofmanyconcretemathematicaladvancesmadebythemethodsofformallogic. DuringthesecondyearofstudiesIdecidedtoauditacourseintheoreticalphysicsgivenbyProfessorWojciechRubinowicz,aleadingPolishtheoreticiananda formerstudentandcollaboratorofthefamousMunichphysicistSommerfeld. Iattendedhismasterlylecturesonelectromagnetismandtookpartinaseminarheledongrouptheoryandquantumtheoryforadvancedstudents.WeusedHermann Weyl'sGruppenTheorieundQuantumMechanik.ItwasimpressivetoseethehighlevelofmathematicsinvolvedinthestudyofMaxwell'sequationsandinthe theoryofelectricitywhichmadeupitsfirstpart.Eventhoughmuchofitwasabovemyheadtechnically,Imanagedtodoalotofreadingontheside.Ireadpopular accountsoftheoretical

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physicsinstatisticalmechanics,inthetheoryofgasesandthetheoryofrelativity,andonelectricityandmagnetism. Duringthewinter,Rubinowiczfellillandaskedme(althoughIwastheyoungestmemberoftheclass)toconductafewsessionsduringhisabsence.Iremembertothis dayhowIstruggledwiththeunfamiliaranddifficultmaterialofWeyl'sbook.Thiswasmyfirstactiveparticipationintheareaofphysics. ThemathematicsofficesofthePolytechnicInstitutecontinuedtobemyhangout.Ispentmorningsthere,everydayoftheweek,includingSaturdays.(Saturdayswere notconsideredtobepartoftheweekendthen;classeswereheldonSaturdaymornings.) Mazurappearedoften,andwestartedouractivecollaborationonproblemsoffunctionspaces.Wefoundasolutiontoaprobleminvolvinginfinitelydimensional vectorspaces.Thetheoremweproved—thatatransformationpreservingdistancesislinear—isnowpartofthestandardtreatmentofthegeometryoffunction spaces.WewroteapaperwhichwaspublishedintheCompte­RendusoftheFrenchAcademy. ItwasMazur(alongwithKuratowskiandBanach)whointroducedmetocertainlargephasesofmathematicalthinkingandapproaches.FromhimIlearnedmuch abouttheattitudesandpsychologyofresearch.Sometimeswewouldsitforhoursinacoffeehouse.Hewouldwritejustonesymboloralinelikey=f(x)onapiece ofpaper,oronthemarbletabletop.Wewouldbothstareatitasvariousthoughtsweresuggestedanddiscussed.Thesesymbolsinfrontofuswerelikeacrystalball tohelpusfocusourconcentration.YearslaterinAmerica,myfriendEverettandIoftenhadsimilarsessions,butinsteadofacoffeehousetheywereheldinanoffice withablackboard. Mazur'sfortewasmakingwhathecalled"observationsandremarks."Thesestated—usuallyinaconciseandpreciseform—somepropertiesofnotions.Oncemade, theywereperhapsnotsodifficulttoverify,forsometimesthey

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wereperipheraltotheusualformulationsandhadgoneunnoticed.Theywereoftendecisiveinsolvingproblems. Inaconversationinthecoffeehouse,Mazurproposedthefirstexamplesofinfinitemathematicalgames.Irememberalso(itmusthavebeensometimein1929or 1930)thatheraisedthequestionoftheexistenceofautomatawhichwouldbeabletoreplicatethemselves,givenasupplyofsomeinertmaterial.Wediscussedthis veryabstractly,andsomeofthethoughtswhichweneverrecordedwereactuallyprecursorsoftheorieslikethatofvonNeumannonabstractautomata.We speculatedfrequentlyaboutthepossibilityofbuildingcomputerswhichcouldperformexploratorynumericaloperationsandevenformalalgebraicalwork. IhavementionedthatIfirstsawBanachataseriesofmathematicslectureswhenIwasinhighschool.Hewastheninhismiddlethirties,butcontrarytotheimpression giventoveryyoungpeoplebymenfifteenortwentyyearstheirsenior,tomeheappearedtobeveryyouthful.Hewastall,blond,blue­eyed,andratherheavy­set.His mannerofspeakingstruckmeasdirect,forceful,andperhapstoosimple­minded(atraitwhichIlaterobservedwastosomeextentconsciouslyforced).Hisfacial expressionwasusuallyoneofgoodhumormixedwithacertainskepticism. Banachcamefromapoorfamily,andhehadverylittleconventionalschoolingatfirst.Hewaslargelyself­taughtwhenhearrivedatthePolytechnicalInstitute.Itis saidthatSteinhausaccidentallydiscoveredhistalentwhenheoverheardamathematicalconversationbetweentwoyoungstudentssittingonaparkbench.Onewas Banach,theotherNikodym,nowrecentlyretiredasprofessorofmathematicsatKenyonCollege.BanachandSteinhausweretobecometheclosestofcollaborators andthefoundersoftheLwówschoolofmathematics. Banach'sknowledgeofmathematicswasbroad.Hiscontributionswereinthetheoryoffunctionsofrealvariables,settheoryand,aboveall,functionalanalysis,the theoryofspacesofinfinitelymanydimensions(thepointsofthese

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spacesbeingfunctionsorinfiniteseriesofnumbers).Theyincludesomeofthemostelegantresults.Heoncetoldmethatasayoungmanheknewthethreevolumesof Darboux'sDifferentialGeometry. IattendedonlyafewofBanach'slectures.Iespeciallyremembersomeonthecalculusofvariations.Ingeneral,hislectureswerenottoowellprepared;hewould occasionallymakemistakesoromissions.Itwasmoststimulatingtowatchhimworkattheblackboardashestruggledandinvariablymanagedtopullthrough.Ihave alwaysfoundsuchalecturemorestimulatingthantheentirelypolishedoneswheremyattentionwouldlapsecompletelyandwouldreviveonlywhenIsensedthatthe lecturerwasindifficulty. Beginningwiththethirdyearof'studies,mostofmymathematicalworkwasreallystartedinconversationswithMazurandBanach.AndaccordingtoBanachsomeof myowncontributionswerecharacterizedbyacertain"strangeness"intheformulationof'problemsandintheoutlineofpossibleproofs.Ashetoldmeoncesome yearslater,hewassurprisedhowoftenthese"strange"approachesreallyworked.Suchastatement,comingfromthegreatmastertoayoungmanoftwenty­eight, wasperhapsthegreatestcomplimentIhavereceived. Inmathematicaldiscussions,orinshortremarkshemadeongeneralsubjects,onecouldfeelalmostatoncethegreatpowerofhismind.Heworkedinperiodsof greatintensityseparatedbystretchesofapparentinactivity.Duringthelatterhismindkeptworkingonselectingthestatements,thesortofalchemist'sprobestones thatwouldbestserveasfocaltheoremsinthenextfieldofstudy. Heenjoyedlongmathematicaldiscussionswithfriendsandstudents.IrecallasessionwithMazurandBanachattheScottishCaféwhichlastedseventeenhours withoutinterruptionexceptformeals.Whatimpressedmemostwasthewayhecoulddiscussmathematics,reasonaboutmathematics,andfindproofsinthese conversations. Sincemanyofthesediscussionstookplaceinneigh­

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borhoodcoffeehousesorlittleinns,somemathematiciansalsodinedtherefrequently.Itseemstomenowthefoodmusthavebeenmediocre,butthedrinkswere plentiful.Thetableshadwhitemarbletopsonwhichonecouldwritewithapencil,and,moreimportant,fromwhichnotescouldbeeasilyerased. Therewouldbebriefspurtsofconversation,afewlineswouldbewrittenonthetable,occasionallaughterwouldcomefromsomeoftheparticipants,followedby longperiodsofsilenceduringwhichwejustdrankcoffeeandstaredvacantlyateachother.Thecaféclientsatneighboringtablesmusthavebeenpuzzledbythese strangedoings.Itissuchpersistenceandhabitofconcentrationwhichsomehowbecomesthemostimportantprerequisitefordoinggenuinelycreativemathematical work. Thinkingveryhardaboutthesameproblemforseveralhourscanproduceaseverefatigue,closetoabreakdown.Ineverreallyexperiencedabreakdown,buthave felt"strangeinside"twoorthreetimesduringmylife.OnceIwasthinkinghardaboutsomemathematicalconstructions,oneaftertheother,andatthesametimetrying tokeepthemallsimultaneouslyinmymindinaveryconsciouseffort.Theconcentrationandmentaleffortputanaddedstrainonmynerves.Suddenlythingsstarted goingroundandround,andIhadtostop. TheselongsessionsinthecaféswithBanach,ormoreoftenwithBanachandMazur,wereprobablyunique.CollaborationwasonascaleandwithanintensityIhave neverseensurpassed,equaledorapproximatedanywhere—exceptperhapsatLosAlamosduringthewaryears. Banachconfidedtomeoncethateversincehisyouthhehadbeenespeciallyinterestedinfindingproofs—thatis,demonstrationsofconjectures.Hehada subconscioussystemforfindinghiddenpaths—thehallmarkofhisspecialgenius. AfterayearortwoBanachtransferredourdailysessions

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fromtheCaféRomatothe"Szkocka"(ScottishCafé)justacrossthestreet.Stozekwasthereeverydayforacoupleofhours,playingchesswithNikliborcand drinkingcoffee.Othermathematicianssurroundedthemandkibitzed. KuratowskiandSteinhausappearedoccasionally.TheyusuallyfrequentedamoregenteelteashopthatboastedthebestpastryinPoland. ItwasdifficulttooutlastoroutdrinkBanachduringthesesessions.Wediscussedproblemsproposedrightthere,oftenwithnosolutionevidentevenafterseveral hoursofthinking.ThenextdayBanachwaslikelytoappearwithseveralsmallsheetsofpapercontainingoutlinesofproofshehadcompletedinthemeantime.Ifthey werenotpolishedorevennotquitecorrect,Mazurwouldfrequentlyputtheminamoresatisfactoryform. Needlesstosaysuchmathematicaldiscussionswereinterspersedwithagreatdealoftalkaboutscienceingeneral(especiallyphysicsandastronomy),university gossip,politics,thestateofaffairsinPoland;or,touseoneofJohnvonNeumann'sfavoriteexpressions,the"restoftheuniverse."Theshadowofcomingevents,of Hitler'sriseinGermany,andthepremonitionofaworldwarloomedominously. Banach'shumorwasironicalandsometimestingedwithpessimism.ForatimehewasdeanoftheFacultyofScienceandhadtoattendvariouscommitteemeetings. Hetriedtoavoidallsuchactivities,asmuchashecould,andoncehetoldme,"Wiemgdzieniebede[IknowwhereIwon'tbe],"hiswayofsayingthathedidnot intendtoattendadullmeeting. Banach'sfacultyforproposingproblemsilluminatingwholesectionsofmathematicaldisciplineswasverygreat,andhispublicationsreflectonlyapartofhis mathematicalpowers.Thediversityofhismathematicalinterestssurpassedthatshowninhispublishedwork.HispersonalinfluenceonothermathematiciansinLwów andinPoland

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wasverystrong.Hestandsoutasoneofthemainfiguresofthisremarkableperiodbetweenthewarswhensomuchmathematicalworkwasaccomplished. Ihavehadnopreciseknowledgeofhislifeandworkfromtheoutbreakofthewartohisprematuredeathinthefallof1945.Fromfragmentsofinformationobtained later,welearnedthathewasstillinLwówduringtheGermanoccupationandinmiserablecircumstances.SurvivingtoseethedefeatofGermany,hediedin1945of lungdisease,probablycancer.Ihadoftenseenhimsmokefourorfivepacksofcigarettesinaday. In1929KuratowskiaskedmetoparticipateinaCongressofMathematiciansfromtheSlavicCountrieswhichwastotakeplaceinWarsaw.Whatsticksinmymind isareceptioninthePalaceofthePresidiumoftheCouncilofMinistersandmytimidityatseeingsomanygreatmathematicians,governmentofficials,andimportant people.Thiswasovercomesomewhatwhenanothermathematician,Aronszajn,whowasfourorfiveyearsolderthanI,said,"Kolego"(thiswasthewayPolish mathematiciansaddressedeachother),"let'sgototheotherroom,thepastryisverygoodthere."(HeisnowaprofessorattheUniversityofKansasinLawrence.) TheLwówsectionofthePolishMathematicalSocietyhelditsmeetingsattheUniversitymostSaturdayevenings.Usuallythreeorfourshortpapersweregivenduring anhourorso,afterwhichmanyoftheparticipantsrepairedtothecoffeehousetocontinuethedebates.SeveraltimesIannouncedbeforehandthatIhadsomeresults tocommunicateatoneofthesesessionswhenmyproofwasnotcomplete.Ifeltconfident,butIwasalsolucky,becauseIfinishedtheproofsbeforeIhadtospeak. IwasnineteenortwentywhenStozekaskedmetobecomesecretaryoftheLwówSection,ajobwhichmainlyrequiredsendingannouncementsofmeetingsand writingupshortabstractsoftalksfortheSociety'sBulletin.Therewas,ofcourse,muchcorrespondencebetweenoursection

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andtheothersectionsinCracow,Poznan,andWilno.ImportantproblemsaroseabouttransferringtheadministrativeseatoftheSocietyfromCracow,theancient Polishroyalcity,toWarsaw,thecapital,wheretheheadquartersoftheSocietywereeventuallylocated.Needlesstosaythistookagreatdealofmaneuveringand politicking. OnedayalettercamefromtheCracowcentersolicitingthesupportoftheLwówsection.ItoldStozek,whowasthepresidentofoursection,''Animportantletter justarrivedthismorning."Hisreply—"Hideitsonohumaneyewilleverseeitagain"—wasagreatshocktomyyouthfulinnocence. ThesecondbigcongressIattendedwasheldinWilnoin1931.IwenttoWilnobytrainviaWarsawwithStozek,Nikliborc,andoneortwoothermathematicians. Theykeptfortifyingthemselveswithsnacksanddrinks,butwhenIpulledoutaflaskofbrandyfrommypocket,Stozekburstintolaughterandsaid,"Hismamagaveit tohimincaseheshouldfeelfaint."ThismademeacutelyawareofhowyoungIwasintheeyesofothers.FormanyyearsIwastheyoungestamongmymathematical friends.ItmakesmemelancholytorealizethatInowhavebecometheoldestinmostgroupsofscientists. Wilnowasamarvelouscity.QuitedifferentfromthecitiesoftheAustrianpartofPoland,itgaveadefinitelyorientalimpression.Thewholecityappearedexotictome andmuchmoreprimitivethanmypartofPoland.Thestreetswerestillpavedwithcobblestones.WhenIpreparedtotakeabathinmyhotelroom,thegigantic bathtubhadnorunningwater.WhenIrangthebellasturdyfellowinRussianbootsappearedwiththreelargebucketsofhotwatertopourintothetub. IvisitedthechurchofSt.Ann,theonewhichNapoleonadmiredsomuchonhiswaytoMoscowthathewantedtomoveittoFrance. ThiswasthefirstandlasttimeIevervisitedWilno.IshouldmentionherethatoneofthemostprominentPolish

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mathematicians,AntoniZygmund,wasaprofessorthereuntilWorldWarII.HeleftviaSwedenin1940tocometotheUnitedStatesandisnowaprofessoratthe UniversityofChicago. AttheCongressIgaveatalkabouttheresultsobtainedwithMazurongeometricalisometrictransformationsofBanachspaces,demonstratingthattheyarelinear. Someoftheadditionalremarkswemadeatthetimearestillunpublished.Ingeneral,theLwówmathematicianswereonthewholesomewhatreluctanttopublish.Was itasortofposeorapsychologicalblock?Idon'tknow.ItespeciallyaffectedBanach,Mazur,andmyself,butnotKuratowski,forexample. Muchofthehistoricaldevelopmentofmathematicshastakenplaceinspecificcenters.Thesecenters,largeorsmall,haveformedaroundasinglepersonorafew individuals,andsometimesasaresultoftheworkofanumberofpeople—agroupinwhichmathematicalactivityflourished.Suchagrouppossessesmorethanjusta communityofinterests;ithasadefinitemoodandcharacterinboththechoiceofinterestsandthemethodofthought.Epistemologicallythismayappearstrange,since mathematicalachievement,whetheranewdefinitionoraninvolvedproofofaproblem,mayseemtobeanentirelyindividualeffort,almostlikeamusicalcomposition. However,thechoiceofcertainareasofinterestisfrequentlytheresultofacommunityofinterests.Suchchoicesareofteninfluencedbytheinterplayofquestionsand answers,whichevolvesmuchmorenaturallyfromtheinterplayofseveralminds.Thegreatnineteenth­centurycenterssuchasGöttingen,Paris,andCambridge (England)allexercisedtheirownpeculiarinfluenceonthedevelopmentofmathematics. TheaccomplishmentsofthemathematiciansinPolandbetweenthetwoworldwarsconstituteanimportantelementinmathematicalactivitythroughouttheworldand havesetthetoneofmathematicalresearchinmanyareas. ThisisdueinparttotheinfluenceofJaniszewski,one

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oftheorganizersofPolishmathematicsandawriteronmathematicaleducation,whounfortunatelydiedyoung.JaniszewskiadvocatedthatthenewstateofPoland specializeinwell­definedareasratherthantrytoworkintoomanyfields.Hisargumentswere,first,thattherewerenotmanypersonsinPolandwhocouldbecome involved,andsecond,thatitwasbettertohaveanumberofpersonsworkinginthesamedomainsotheycouldhavecommoninterestsandcouldstimulateeachother indiscussions.Ontheotherhand,thisreducedsomewhatthescopeandbreadthoftheinvestigations. AlthoughLwówwasaremarkablecenterformathematics,thenumberofprofessorsbothattheInstituteandattheUniversitywasextremelylimitedandtheirsalaries wereverysmall.PeoplelikeSchauderhadtoteachinhighschoolinordertosupplementameagerincomeaslecturerorassistant.(Schauderwasmurderedbythe Germansin1943.)ZbigniewLomnickiworkedasanexpertinprobabilitytheoryinagovernmentinstituteofstatisticsandinsurance.IfIhadtonameonequality whichcharacterizedthedevelopmentofthisschool,madeupofthemathematiciansfromtheUniversityandthePolytechnicInstitute,Iwouldsaythatitwastheir preoccupationwiththeheartofthematterthatformsmathematics.BythisImeanthatifoneconsidersmathematicsasresemblingatree,theLwówgroupwasintent onthestudyoftherootsandthetrunkratherthanthebranches,twigs,andleaves.Onasettheoreticalandaxiomaticbasisweexaminedthenatureofageneralspace, thegeneralmeaningofcontinuity,generalsetsofpointsinEuclideanspace,generalfunctionsofrealvariables,ageneralstudyofthespacesoffunctions,ageneralidea ofthenotionsoflength,areaandvolume,thatistosay,theconceptofmeasureandtheformulationofwhatshouldbecalledprobability. Inretrospectitseemssomewhatcuriousthattheideasofalgebrawerenotconsideredinasimilargeneralsetting.Itisequallycuriousthatstudiesofthefoundationsof phy­

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sics—inparticularastudyofspace­time—havenotbeenundertakeninsuchaspiritanywheretothisday. Lwówhadfrequentandlivelyinteractionwithothermathematicalcenters,especiallyWarsaw.FromWarsawSierpinskiwouldcomeoccasionally,sowould Mazurkiewicz,Knaster,andTarski.InLwówtheywouldgiveshorttalksatthemeetingsofthemathematicalsocietyonSaturdayevenings.Sierpinskiespeciallyliked theinformalLwówatmosphere,theexcursionstoinnsandtaverns,andthegaydrinkingwithBanach,Ruziewicz,andothers.(Ruziewiczwasmurderedbythe GermansonJuly4,1941.) MazurkiewiczoncespentasemesterlecturinginLwów.LikeKnasterintopology,hewasamasteratfindingcounterexamplesinanalysis,examplesshowingthata conjectureisnottrue.Hiscounterexamplesweresometimesverycomplicated,butalwaysingeniousandelegant. Sierpinski,withhissteadystreamofresultsinabstractsettheoryorinsettheoreticaltopology,wasalwayseagertolistentonewproblems—evenminorones—andto thinkaboutthemseriously.OftenhewouldsendsolutionsbackfromWarsaw. BronislawKnasterwastall,bald,veryslim,withflashingdarkeyes.HeandKuratowskipublishedmanypaperstogether.Hewasreallyanamateurmathematician, veryingeniousattheconstructionofsetsofpointsandcontinuawithpathologicalproperties.HehadstudiedmedicineinParisduringthefirstWorldWar.Being extremelywitty,heusedtoentertainuswithdescriptionsofthepolyglotinternationalgroupofstudentsandtheindescribablelanguagetheyspoke.Hequotedone studenthehadoverheardinarestaurantashavingsaid:"Kolego,pozaluitemniaeinstückelevondiesemfaschiertenpoisson,"anamalgamofPolish,Russian,Yiddish, German,andFrench! Borsuk,moremycontemporary,cameforalongervisitfromWarsaw.Westartedcollaboratingfromthefirst.FromhimIlearnedaboutthetrulygeometric,more visual,almost

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"palpable"tricksandmethodsoftopology.OurresultswerepublishedinanumberofpaperswhichwesenttoPolishjournalsandtosomejournalsabroad.Actually myfirstpublicationintheUnitedStatesappearedwhileIwasinLwów.ItwasajointpaperwithBorsuk,publishedintheBulletinoftheAmericanMathematical Society.Wedefinedtheideaof"epsilonhomeomorphisms"—approximatehomeomorphisms—andthebehaviorofsometopologicalinvariantsundersuchmore generaltransformation's—continuousones,butnotnecessarilyonetoone.Ajointpaperonsymmetricproductsintroducedanideathatmodifiesthedefinitionofa Cartesianproductandleadstotheconstructionofsomecuriousmanifolds.Someofthesemightonedayfindapplicationsinphysicaltheories.Theycorrespondtothe newstatisticsofcountingthenumbersofparticles(notinthefamiliarclassicalsense,butratherinthespiritofquantumtheorystatisticsofindistinguishableparticles,or ofparticlesobeyingtheBose­EinsteinorelseFermi­Diracwaysofcountingtheircombinationsanddispositions).Thesecannotbeexplainedhere;perhapsthis mentionwillwhetthecuriosityofsomereaders. KuratowskiandSteinhaus,eachinadifferentway,representedelegance,rigor,andintelligenceinmathematics.KuratowskiwasreallyarepresentativeoftheWarsaw schoolwhichflourishedalmostexplosivelyafter1920.HecametoLwówin1927,precededbyareputationforhisworkinpuresettheoryandaxiomatictopologyof generalspaces.AseditorofFundamentaMathematicaeheorganizedandgavedirectiontomuchoftheresearchinthisfamousjournal.Hismathematicswas characterizedbywhatIwouldcallaLatinclarity.Intheproliferationofmathematicaldefinitionsandinterests(nowevenmorebewilderingthanatthattime), Kuratowski'smeasuredchoiceofproblemshadthequalityofwhatishardtodefine—commonsenseintheabstractions. SteinhauswasoneofthefewPolishprofessorsofJewishdescent.Hecamefromawell­known,quiteassimilated

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Jewishfamily.Acousinofhishadbeenagreatpatriot,oneofthePilsudskilegionnaires;hewaskilledduringthefirstWorldWar. Steinhaus'ssenseofanalysis,hisfeelingsforproblemsinrealvariables,infunctiontheory,inorthogonalseriesmanifestedagreatknowledgeofhistoricaldevelopment ofmathematicsandcontinuityofideas.Perhapswithoutsomuchinterestorfeelingfortheveryabstractpartsofmathematics,healsosteeredsomenewmathematical ideasinthedirectionofpracticalapplications. Hehadatalentforapplyingmathematicalformulationstomattersascommonasproblemsofdailylife.Certainlyhisinclinationsweretosingleoutproblemsof geometrythatcouldbetreatedfromacombinatorialpointofview—actuallyanythingthatpresentedthevisual,palpablechallengeofamathematicaltreatment. Hehadgreatfeelingforlinguistics,almostpedanticattimes.Hewouldinsistonabsolutelycorrectlanguagewhentreatingmathematicsordomainsofscience susceptibletomathematicalanalysis. Auerbachwasrathershort,stooped,andusuallywalkedwithhisheaddown.Outwardlytimid,hewasoftencapableofverycaustichumor.Hisknowledgeof classicalmathematicswasprobablygreaterthanthatofmostoftheotherprofessors.Forexample,heknewclassicalalgebraverywell. AthisinstigationMazur,afewothers,andIdecidedtostartasystematicstudyofLiegroupsandothertheorieswhichwerenotstrictlyinthedomainofwhatisnow calledPolishmathematics.Auerbachalsoknewalotaboutgeometry.Ihadmanydiscussionswithhimonthetheoryofconvexbodies,towhichMazurandI contributedseveraljointpapers. AuerbachandIplayedchessattheCaféRomaandoftenwentthroughthefollowinglittleritualwhenIbeganwithacertainopening(atthattimeIdidnotknowany theoryofchessopeningsandplayedbyintuitiononly).WhenImadethosemoveswiththekingpawnhewouldsay,"Ah!Ruy

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Lopez."Iwouldask,"Whatisthat?"andhewouldreply,"ASpanishbishop." Auerbachdiedduringthewar.IunderstandthatheandSternbachtookpoisonwhilebeingtransportedbytheGermanstoaninterrogationsession,butIdonotknow thecircumstancesof'theirarrestoranythingelseabouttheirlivesbeforeandduringtheNazioccupation. IbelievemycollaborationwithSchreierstartedwhenIwasinmysecondyearofstudies.OfthemathematiciansattheUniversityandatthePolytechnicInstitute,he wastheonlyonewhowasmorestrictlymycontemporary,sincehewasonlysixmonthsorayearolderandstillastudentattheUniversity.Wemetinaseminarroom duringalecturebySteinhausandtalkedaboutaproblemonwhichIwasworking.Almostimmediatelywefoundmanycommoninterestsandbegantoseeeachother regularly.Awholeseriesofpaperswhichwewrotejointlycamefromthiscollaboration. Wewouldmeetalmosteveryday,occasionallyatthecoffeehousebutmoreoftenatmyhouse.HishomewasinDrohobycz,alittletownandpetroleumcentersouth ofLwów.Whatavarietyofproblemsandmethodswediscussedtogether!Ourwork,whilestillinspiredbythemethodsthencurrentinLwów,branchedintonew fields:groupsoftopologicaltransformations,groupsofpermutations,puresettheory,generalalgebra.Ibelievethatsomeofourpaperswereamongthefirsttoshow applicationstoawiderclassofmathematicalobjectsofmodernsettheoreticalmethodscombinedwithamorealgebraicpointofview.Westartedworkonthetheory ofgroupoids,aswecalledthem,orsemi­groups,astheyarecallednow.Severaloftheseresultscanbefoundintheliteraturebynow,butsomeothershavenotyet appearedinprintanywheretomyknowledge. SchreierwasmurderedbytheGermansinDrohobyczinApril,1943. Anothermathematician,MarkKac,fourorfiveyearsmyjunior,wasastudentofSteinhaus.Asabeginningunder­

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graduatehehadalreadyshownexceptionaltalent.MyconnectionswithKacdevelopedalittlelaterduringmysummervisitstoLwów,whenIbegantospend academicyearsatHarvard.HealsohadthegoodfortunetocometotheUnitedStates,afewyearsafterIdid,andourfriendshipstartedinfullmeasureonlyinthis country. In1932IwasinvitedtogiveashortcommunicationattheInternationalMathematicalCongressinZürich.ThiswasthefirstbiginternationalmeetingIattended,andI feltveryproudtohavebeeninvited.IncontrasttosomeofthePolishmathematiciansIknew,whowereterriblyimpressedbywesternscience,Ihadconfidenceinthe equalvalueofPolishmathematics.Actuallythisconfidenceextendedtomyownwork.VonNeumannoncetoldmywife,Françoise,thathehadnevermetanyone withasmuchself­confidence—addingthatperhapsitwassomewhatjustified. Travelingwest,IfirstjoinedKuratowski,Sierpinski,andKnasterinVienna.TheyhadallcomefromKuratowski'ssummerplacenearWarsaw;onthewaytoZürich theprofessorsdecidedtostopinInnsbrück.WemetsomemathematiciansfromothercountriesalsoontheirwaytotheCongressandspentacoupleofdaysthere.I rememberanexcursionbycablecartoamountaincalledHafelekar.ThiswasthefirsttimeIwaseverabovetwothousandmeters,andtheviewwasbeautiful.I rememberfeelingalittledizzyforafewminutesandidentifyingthisfeelingwithoneIhadhadpreviouslyonseveraloccasionswhengettingthesalientpointsofproofs oftheoremsIstudiedinhighschool. TheCongressinZürichwasanenormousaffaircomparedtoanyIhadpreviouslyattended,butquitesmallincomparisontothoseafterWorldWarII.Istillhavea photographofallthemembersstandinginfrontoftheTechnischeHochschule.ThereforthefirsttimeIsawandmetmanyforeignmathematicians. Themeetingwasinteresting,andIfounditstimulatingtohearaboutmanytypesorfieldsofmathematicsotherthan

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theonescultivatedinPoland.Thediversityofmathematicalfieldsopenednewvistasandsuggestednewideastome.InthosedaysIwenttoalmosteveryavailable generaltalk. ManyoftheGermanandWestEuropeanmathematiciansappearedtomenervous;somehadfacialtwitches.OnthewholecomparedtothePolesIknewthey seemedlessatease.AndeventhoughinPolandtherewasgreatadmirationfortheGöttingenschoolofmathematics,Iagainfelt,perhapsnotjustifiably,myownsense ofself­confidence. Igavemyownlittletalkfeelingonlymoderatelynervous.Thereasonforthiscomparativelackofnervousness,Ithink,inretrospect,wasduetomyattitude, compoundedofacertaindrunkennesswithmathematicsandaconstantpreoccupationwithit. Somebodypointedoutashortoldman.ItwasHilbert.ImettheoldPolishmathematicianDickstein,whowasinhisninetiesandwalkingaroundlookingforhis contemporaries.Dickstein'steacherhadbeenastudentofCauchyintheearlynineteenthcentury,andhestillconsideredPoincaré,whodiedin1912,abrightyoung man.Tomethiswaslikegoingintotheprehistoryofmathematicsanditfilledmewithakindofphilosophicalawe.ImetmyfirstAmericanmathematician,Norbert Wiener.VonNeumannwasnotthere,andthiswasadisappointment.IhadheardsomuchabouthisvisittoLwówin1929. AtthehotelswimmingpoolImetthefamousphysicistPauliwithProfessorWavreandAdaHalpern.Wavre,Ada'sprofessor,wasaSwissmathematician,knownfor hisstudiesofthecelebratedclassicalproblemoffiguresofequilibriumofrotatingplanetaryandstellarbodies,amongotherthings.AdacamefromLwów.Shewasa verygood­lookinggirlwhowasstudyingmathematicsattheUniversityofGeneva.ForafewyearsIhadanoff­and­onromancewithher.Infrontofallthiscompany, IturnedtoPauliandtriedapun,saying:"ThisisaPauliVerbot"(aPaulianphysicalprinciplewhichassertsthattwoparticleswiththesame

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characteristicscannotoccupythesameplace),referringtoWavreandmewhowereboththereinthecompanyofthisprettyyounglady. AnotherinterestingencounteroccurredoneafternooninthewoodsaroundthefamousDolderHotel.Havinglostmyway,IranintoPaulAlexandroffandEmmy Noetherwalkingtogetheranddiscussingmathematics.AlexandroffknewaboutsomeofmyworkforIhadsenthimreprintsandwehadhadsomeprevious mathematicalcorrespondence.InfactoneofthegreatjoysofmylifehadbeentoreceivealetterfromhimaddressedtoProfessorS.Ulam.Duringthisencounterhe suddenlysaidtome:"Ulam,wouldyouliketocometoRussia?Icouldarrangeeverythingandwouldlikeverymuchtohaveyou."AsaPole,andwithmyrather capitalisticfamilybackground,hisinvitationflatteredme,butsuchatripappearedquiteunthinkable. TheCongressover,afteralittleexcursiontoMontreuxwithKuratowskiandKnasterIreturnedtoPolandintimetotakemyMaster'sdegree. Ihadanalmostpathologicalaversiontoexaminations.ForovertwoyearsIhadneglectedtotaketheexaminationswhichwereusuallynecessarytoprogressfromone yeartothenext.Myprofessorshadbeentolerant,knowingthatIwaswritingoriginalpapers.Finally,Ihadtotakethem—allatonce. Istudiedforafewmonths,tookakindofcomprehensiveexaminationandwrotemyMaster'sthesisonasubjectwhichIthoughtupmyself.Iworkedforaweekon thethesis,thenwroteitupinonenight,fromabouttenintheeveninguntilfourinthemorning,onmyfather'slongsheetsoflegalpaper.Istillhavetheoriginal manuscript.(Itisunpublishedtothisday.)Thepapercontainsgeneralideasontheoperationsofproductsofsets,andsomeofitoutlineswhatisnowcalledCategory Theory.Italsocontainssomeindividualresultstreatingveryabstractlytheideaofageneraltheoryofmanyvariablesindiversepartsofmath­

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ematics.Allthiswasinthefallof1932uponmyreturnfromZürich. In1933ItookmyDoctor'sexamination.ThethesiswaspublishedbyOssolineum,anestablishmentwhichprintedtheLwówperiodicalStudiaMathematica.It combinedseveralofmyearlierpapers,theorems,andgeneralizationsinmeasuretheory. MydegreewasthefirstdoctorateawardedatthePolytechnicInstituteinLwówfromthenewDepartmentofGeneralStudieswhichhadbeenestablishedin1927.It wastheonlydepartmentthatgaveMaster'sandDoctor'sdegrees,alltheothersbeingengineeringdegrees. Theceremonywasaratherformalaffair.IttookplaceinalargeInstitutehallwithfamilyandfriendsattending.Ihadtowearawhitetieandgloves.Mysponsors StozekandKuratowskieachgavealittlespeechdescribingmyworkandthepapersIhadwritten.Afterafewwordsaboutthethesis,theyhandedmeaparchment document. The''aula"—thelargehallinwhichtheceremonytookplace—wasdecoratedwithtraditionalfrescoes.ThesewereverymuchlikesomeIsawtwentyyearslateron thewallsoftheMITcafeteria.TheMITfrescoesdepictscantilydressedwomeninposturesofflight,symbolizingsciencesandarts,andalargefemalefigureofa goddesshoveringoverarecoilingoldman.IusedtojokethatitrepresentedtheAirForcegivingacontracttophysicistsandmathematicians.InFuldHall,theInstitute BuildinginPrinceton,thereisalsoanoldpaintinginthetearoomwherepeopleassembleforconversationintheafternoon.Thereagainoneseesanoldmanwho seemstobeshyingawayfromanangelcomingdownfromtheclouds.WhenIwastoldthatnobodyknewwhatitwassupposedtorepresent,Isuggestedthatitmight bearepresentationofMinnaRies,theladymathematicianwhodirectedtheOfficeofNavalResearchatthetime,proposingaNavycontracttoEinstein,whois recoilinginhorror.

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AftertheexaminationsandceremoniesIpublishedafewmorepapersandthenhadtotakeiteasyfortherestof1933,forabadparatyphoidinfectionleftmeweak forseveralmonths—oneoftheraretimesinmylifewhenIwasseriouslyill. Butnotallwasseriousworkandnoplay.Intheearly1930s,ahighschoolteacherofsciencebythenameofHirniak,awizened,smallman,cametoourcoffee house.Hewouldsitafewtablesawayfromus,sippingvodkaandcoffeeinturn,andbusilyscribblingonapadofpaper.Everyonceinawhilehewouldgetupand joinourtabletogossiporkibitzwhenNikliborcandStozekplayedchess.Nikliborcwouldrepeatwithglee:"Gehirn[braininGerman],Gehirniak!" Hirniak,whotaughtmathematics,physics,andchemistry,wastryingtosolveFermat'sfamousproblem.Thisisoneofthebest­knownunsolvedproblemsin mathematics,andforalongtimehasattractedcranksaswellasamateurs,whoregularlyproducefalseorveryincompleteproofsofFermat'sconjecture. Hirniakwasafixtureatthecoffeehouse,andhisconversationwasdelightfullypicturesqueandfullofunconsciouslyhumorousstatements.Wewouldcollectand repeatthemtoeachother;Iusedtopastesomeofthemonthewallsofmyroomathome. ItturnedoutthatmyfatherknewHirniak,whosewifeownedalargesoda­waterfactoryandwhoselegalaffairswerehandledbymyfather'sfirm.Myfather consideredHirniakahumorouslyfoolishperson.WhenhesawmycollectionofHirniakmaxims,Ibelievehewassurprisedandperhapsevenwonderedaboutmy sanity.Ihadtoexplaintohimthesubtletyofthehumoranditsspecialappealtomathematicians. HirniakwouldtellBanach,forinstance,thattherewerestillsomegapsinhisproofofFermat'sproblem.Thenhewouldadd,"Thebiggermyproof,thesmallerthe hole.Thelongerandlargertheproof,thesmallerthehole."Toa

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mathematicianthisconstitutesanamusingformulation.Hewouldmakeweirdstatementsaboutphysics.Forexample,hewouldsaythathalftheelementsinthe periodictablearemetalsandtheotherhalfarenot.Whensomeonepointedoutthatthiswasriotquitecorrect,hewouldreply:"Ah,butbydefinitionwecancallafew moreofthemmetals!"Hehadawonderfulwayoftakinglibertieswithdefinitions. HestudiedinGöttingenanddescribedhowhewoulddrinkcupsofwinefromanautomaticdispenser.Oncesomethingwentwronginthemachine,andthewine continuedtoflow.Hirniakcontinuedtodrinkuntilhefoundhimselflyingonthegroundsurroundedbyagroupofpeople.Heheardsomeoneask,"Vielleichtistetwas los?"(Maybesomethingwentwrong?).Hereplied,"Vielleichtnicht."(Maybenot.)Atthis,hewascarriedhomeintriumphontheshouldersofthecrowd. HereisthestorythatentertainedvonNeumannsomuchwhenItoldhimaboutHirniakyearslaterinPrinceton:onedayHirniaktoldBanach,Mazur,andmethathe hadalmostprovedFermat'sconjectureandthatAmericanreporterswouldfindoutaboutitandwouldcometoLwówandsay:"Whereisthisgenius?Givehimone hundredthousanddollars!"AndBanachwouldecho:"Giveittohim!"Afterthewar,JohnnysaidtomeonedayinLosAlamos,"Rememberhowweusedtolaughat Hirniak'shundredthousanddollarsstory?Well,hewasright,hewastherealprophetwhilewewerelaughinglikefools."WhatJohnnywasreferringto,ofcourse,was thatrepresentativesoftheDefenseDepartment,theAirForce,andtheNavyweretravelingaroundtheCountryatthetimebountifullydispensingresearchcontractsto scientists.Theaveragecontractamountedtoaboutonehundredthousanddollars.''Notonlywasheright,"saidJohnny,"butheevenforesawthecorrectamount!" Sometimearound1933or1934,BanachbroughtintotheScottishCaféalargenotebooksothatwecouldwritestatementsofnewproblemsandsomeoftheresults ofourdis­

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cussionsinmoredurableform.Thisbookwaskepttherepermanently.Awaiterwouldbringitondemandandwewouldwritedownproblemsandcomments,after whichthewaiterwouldceremoniouslytakeitbacktoitssecretcache.Thisnotebookwaslatertobecomefamousas"TheScottishBook." Manyoftheproblemsdatefrombefore1935.Theywerediscussedagreatdealbythosewhosenameswereincluded.Mostofthequestionsposedweresupposed tohavereceivedconsiderableattentionbeforean"official"inclusioncouldbeconsidered.Inseveralcases,theproblemsweresolvedonthespotandtheanswers included. ThecityofLwówandtheScottishBookwerefatedtohaveaverystormyhistorywithinafewyearsofthebook'sinception.AftertheoutbreakofWorldWarII,the citywasoccupiedbytheRussians.FromitemstowardtheendofthebookitisevidentthatsomeRussianmathematiciansmusthavevisitedthetown.Theyleftseveral problemsandoffersofprizesfortheirsolution.ThelastdateappearinginthebookisMay31,1941.ItemNo.193containsarathercrypticsetofnumericalresults signedbySteinhausdealingwiththedistributionofthenumberofmatchesinabox!AfterthestartofthewarbetweenGermanyandRussia,thecitywasoccupiedby Germantroopsinthesummerof1941,andthenotesceased.Thefateofthebookduringtheremainingyearsofthewarisnotknowntome.AccordingtoSteinhaus, thisdocumentwasbroughttoWroclaw(formerlyBreslau)byBanach'sson,nowaneurosurgeoninPoland. DuringmylastvisittoLwówinthesummerof1939,afewdaysbeforeIleftIhadaconversationwithMazuronthelikelihoodofwar.Peoplewereexpectinganother crisislikeMunichandwerenotpreparedforanimminentworldwar.Mazursaidtome,"Aworldwarmaybreakout.WhatshallwedowiththeScottishBookand ourjointunpublishedpapers?YouareleavingfortheUnitedStatesandpresumablywillbesafe.Incaseofabombardmentofthecity,Ishallputthemanuscriptsand thebookinacase,

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whichIshallburyintheground."Weevendecidedonalocation.Itwastobenearthegoalpostofafootballfieldjustoutsidethecity.Idonotknowwhetheranyof thisreallyhappened,butapparentlythemanuscriptoftheScottishBooksurvivedingoodshape,forSteinhaussentmeacopyofitafterthewar.Itranslateditin1957 anddistributedittomanymathematicalfriendsintheUnitedStatesandabroad. OfthesurvivingmathematiciansfromLwówmanyarecontinuingtheirworktodayinWroclaw.ThetraditionoftheScottishBookcontinues.Since1945new problemshavebeenposedandrecordedandanewvolumeisinprogress.

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Chapter3 TravelsAbroad 1934 By1934Ihadbecomeamathematicianratherthananelectricalengineer.ItwasnotsomuchthatIwasdoingmathematics,butratherthatmathematicshadtaken possessionofme.Perhapsthisisagoodplacetostopforamomentandponderwhatbeingamathematicianmeans. Theworldofmathematicsisacreationofthebrainandcanbevisualizedwithoutexternalhelp.Mathematiciansareabletoworkontheirsubjectwithoutanyofthe equipmentorpropsneededbyotherscientists.Physicists(eventheoreticalphysicists),biologists,andchemistsneedlaboratories—butmathematicianscanwork withoutchalkorpencilandpaper,andtheycancontinuetothinkwhilewalking,eating,eventalking.Thismayexplainwhysomanymathematiciansappearturned inwardorpreoccupiedwhileperformingotheractivities.Thisisquitepronouncedandquantitativelydifferentfromthebehaviorofscientistsinotherfields.Ofcourse,it dependsontheindividual.Some,

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likePaulErdös,havethischaracteristicintheextreme.Hispreoccupationwithmathematicalconstructionorreasoningoccupiesaverylargepercentageofhiswaking hours,totheexclusionofeverythingelse. Asformyself,eversinceIstartedlearningmathematicsIwouldsaythatIhavespent—regardlessofanyotheractivity—ontheaveragetwotothreehoursaday thinkingandtwotothreehoursreadingorconversingaboutmathematics.SometimeswhenIwastwenty­threeIwouldthinkaboutthesameproblemwithincredible intensityforseveralhourswithoutusingpaperorpencil.(Bytheway,thisisinfinitelymorestrenuousthanmakingcalculationswithsymbolstolookatandmanipulate.) Onthewhole,Istillfindconversationwithorlisteningtootherpeopleaneasierandpleasanterwayoflearningthanreading.TothisdayIcannotread"howto" instructionsinprintedform.Psychologically,theseareindigestibleforme. Somepeopleprefertolearnlanguagesbytherulesofgrammarratherthanbyear.Thiscanbesaidtobetrueofmathematics—somelearnitby"grammar"andothers "fromtheair."Ilearnedmymathematicsfromtheair. Forexample,Ilearned,subconsciously,fromMazurhowtocontrolmyinbornoptimismandhowtoverifydetails.Ilearnedtogomoreslowlyoverintermediatesteps withaskepticalmindandnottoletmyselfbecarriedaway.Temperament,generalcharacter,and"hormonal"factorsmustplayaveryimportantroleinwhatis consideredtobeapurely"mental"activity."Nervous"characteristicsplayanenormousroleinone'sintellectualdevelopment.Bytheageofabouttwenty,when developmentissupposedtobefullycompleted,someoftheseacquiredtraitsareperhapsessentiallyfrozenandhavebecomeapermanentpartofourmakeup. Mathematicsissupposedtobeinessenceonlyaverygeneralandpreciselanguage,butperhapsthisisonlypartiallytrue.Therearemanywaysofexpressingoneself. A

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personwhostartsearlyhassomeparticularwayoforganizinghismemoryordeviseshisownparticularsystemforarrangingimpressions.A"subconscious brewing"(orpondering)sometimesproducesbetterresultsthanforced,systematicthinking,aswhenplanninganoverallprogramincontrasttopursuingaspecificline ofreasoning.Forcingoneselftopersistinalogicalexplorationbecomesahabit,afterwhichitceasestobeforcingsinceitcomesautomatically(asasubroutine,as computerpeopleliketosay).Also,evenifonecannotdefinewhatwecalloriginality,itmighttosomeextentconsistofamethodicalwayofexploringavenues—an almostautomaticsortingofattempts,acertainpercentageofwhichwillbesuccessful. Ialwayspreferredtotrytoimaginenewpossibilitiesratherthanmerelytofollowspecificlinesofreasoningormakeconcretecalculations.Somemathematicianshave thistraittoagreaterextentthanothers.Butimaginingnewpossibilitiesismoretryingthanpursuingmathematicalcalculationsandcannotbecontinuedfortoolonga time. Anindividual'soutputis,ofcourse,conditionedbywhathecanaccomplishmosteasilyandthisperhapsrestrictsitsscope.InmyselfInoticeahabitoftwistinga problemaround,seekingthepointwherethedifficultymaylie.Mostmathematiciansbegintoworrywhentherearenomoredifficultiesorobstaclesfor"newtroubles." Needlesstosay,somedoitmoreimaginativelythanothers.PaulErdösconcentratesallthetime,butusuallyonlineswhicharealreadybegunorwhichareconnected towhathewasthinkingaboutearlier.Hedoesn'twipehismemorycleanlikeataperecordertostartsomethingnew. Banachusedtosay,"Hopeisthemotheroffools,"aPolishproverb.Nevertheless,itisgoodtobehopefulandbelievethatwithluckonewillsucceed.Ifoneinsists onlyoncompletesolutionstoproblems,thisislessrewardingthanrepeatedtrieswhichresultinpartialanswersoratleastinsomeexperience.Itisanalogousto exploringanunknown

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countrywhereonedoesnotimmediatelyhavetoreachtheendofthetrailorallthesummitstodiscovernewrealms. Itismostimportantincreativesciencenottogiveup.Ifyouareanoptimistyouwillbewillingto"try"morethanifyouareapessimist.Itisthesameingameslike chess.Areallygoodchessplayertendstobelieve(sometimesmistakenly)thatheholdsabetterpositionthanhisopponent.This,ofcourse,helpstokeepthegame movinganddoesnotincreasethefatiguethatself­doubtengenders.Physicalandmentalstaminaareofcrucialimportanceinchessandalsoincreativescientificwork. Itiseasiertoavoidmistakesinthelatter,inthatonecancomebacktorethinking;inchessoneisnotallowedtoreconsidermovesoncetheyhavebeenmade. Theabilitytoconcentrateandthedecreaseinawarenessofone'ssurroundingscomemorenaturallytotheyoung.Mathematicianscanstartveryyoung,insomecases intheirteens.InEurope,evenmorethaninAmerica,mathematiciansexhibitprecocity,educationinEuropeanhighschoolshavingbeenseveralyearsaheadofthe moretheoreticaleducationintheUnitedStates.Itisnotunusualformathematicianstoachievetheirbestresultsatanearlyage.Therearesomeexceptions;for instance,Weierstrass,whowasahigh­schoolteacher,achievedhisbestresultswhenhewasforty.Morerecently,NormanLevinsonprovedaverybeautifultheorem whenhewassixty­oneorsixty­two. Attwenty­five,Ihadestablishedsomeresultsinmeasuretheorywhichsoonbecamewellknown.Thesesolvedcertainsettheoreticalproblemsattackedearlierby Hausdorff,Banach,Kuratowski,andothers.ThesemeasureproblemsagainbecamesignificantyearslaterinconnectionwiththeworkofGödelandmorerecently withthatofPaulCohen.Iwasalsoworkingintopology,grouptheory,andprobabilitytheory.FromthebeginningIdidnotbecometoospecialized.AlthoughIwas doingalotofmathematics,Ineverreallyconsideredmyselfasonlyamathematician.Thismay

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beonereasonwhyinlaterlifeIbecameinvolvedinothersciences. In1934,theinternationalsituationwasbecomingominous.HitlerhadcometopowerinGermany.HisinfluencewasfeltindirectlyinPoland.Therewereincreasing displaysofinflamednationalism,extremerightistoutbreaksandanti­Semiticdemonstrations. Ididnotconsciouslyrecognizetheseportentsofthingstocome,butfeltvaguelythatifIwasgoingtoearnalivingbymyselfandnotcontinueindefinitelytobe supportedbymyfather,Imustgoabroad.ForyearsmyuncleKarolAuerbachhadbeentellingme:"Learnforeignlanguages!"Anotheruncle,MichaelUlam,an architect,urgedmetotryacareerabroad.Formyself,unconsciousasIwasoftherealitiesofthesituationinEurope,Iwaspromptedtoarrangealongishtripabroad mainlybyanurgetomeetothermathematicians,todiscussproblemswiththemand,inmyextremeself­confidence,trytoimpresstheworldwithsomenewresults. Myparentswerewillingtofinancethetrip. Myplansweretogowest(gowest,youngman!);firstIwantedtospendafewweeksinViennatoseeKarlMenger,afamousgeometerandtopologist,whomIhad metinPolandthroughKuratowski.Thiswasthefallof1934,rightaftertheassassinationoftheAustrianPremierDollfuss.Viennawasinastateofupheaval,butIwas soabsorbedandalmostperpetuallydrunkwithmathematicsthatIwasnotreallyawareofit. AfteracoupleofdaysinaViennahotel,ImovedtoaprivateboardinghouseneartheUniversity,whereawidowedladyrentedroomstostudents.Thiswasquitea commonarrangementinthosedays.ThehousewasonalittlestreetnamedafterBoltzmann,agreatphysicistofthenineteenthcentury,oneoftheprincipalcreatorsof thekinetictheoryofgasesandofthermodynamics. IvisitedMengerandathishousemetabrilliantyoungSpanishtopologistnamedJimenezyFlores,whohadalreadysomeniceresultstohiscredit.Wetalked mathematics

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agooddeal.Heseemedverywellknowninnightclubsandintroducedmetothelifeofayoungmanabouttown. FromViennaItraveledtoZürichtomeetHeinzHopf,thetopologist.HewasaprofessoratthefamousTechnischeHochschule,withwhomIhadcorresponded. Hopfknewsomethingaboutmytopologicalresultsandinvitedmetovisittheinstitutetogivetwolectures.OnewasaboutworkIhaddonejointlywithBorsukonthe "antipodaltheorem,"atopologicalproblem.IspokeinGerman,inalectureroomofthedepartmentofagriculture.Irecallthereweremanypicturesofprizecows alongthewalls,whichseemedtolookatmewithsadnessandcommiseration. Nevertheless,thisvisittoZürichwasquitefruitful.IalsometaphysicistnamedGrossman,whowasafewyearsmyseniorandwidelytraveled.Herecommended hotelsinFranceandinEnglandtosuitmypurse.Wediscussedphilosophyandtheroleofmathematicsinphysics. AftertwoweeksinZürich,IwenttoParisforfiveweeks,andthatwassheerdelight.IhadbeeninFrancebefore,butthiswasmyfirstvisittoParis.Myuncle Michael'swifehappenedtolivethereatthetimeandshekindlyofferedtoreceivemeandtosendtomymodesthotelherchauffeuredlimousinetotakeme sightseeing.IwassoembarrassedatthethoughtofbeingseenarrivinginaRollsRoyceoraDusenbergattheLouvreorsomeothermuseum,itfeltsoincongruous, thatIdeclinedheroffer. IwenttotheInstitutPoincaréwithaletterofintroductionfromoneofmyprofessorstothefamousoldmathematicianElieCartan.WhenIenteredhisofficeIplunged directlyintoamathematicaldiscussion,tellinghimhowIhadanideaforasimpleandgeneralproofforsolvingHilbert'sfifthproblemoncontinuousgroups.Atfirsthe saidhedidnotquitefollowmyreasoning,butthenheadded,"Ah!Iseenowwhatyouwanttodo."Cartan'slittlewhitegoatee,vivacioussmile,andsparklingeyes gavehimanappearancewhichsomehowfittedmymentalimageofallFrenchmathematicians.Hewasremarkableformanyreasons,notthe

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leastbecausehehaddonesomeofhisbestworkinhisfifties,whenthecreativityofmostmathematiciansisonthedecline. IattendedseveralseminarsandtalksattheInstitutPoincaréandattheSorbonne.Atthefirstseminar,ayoungFrenchmannamedDePosselhappenedtobetalking aboutoneofmyresults.Itmademeswellwithpride.(DePosselisstillteachinginParis.)IwasinvitedtogiveatalkinasallenamedafterthemathematicianHermite, anotherinthesalleDarboux.ThesehallsandstreetssuchasRueLaplace,RueMonge,RueEuler,visiblesignsthattheabstractionsworkedonbymathematicians weresomehowappreciated,becameheadywineandaddedtomygeneralstateofeuphoria.InmyyouthfulwayIwondered,"Ifonlysomedayahundredyearsfrom now,alittlestreetorevenanalleycouldbenamedafterme." InOctoberIdecidedtogotoCambridge,England.SteinhaushadgivenmealetterofintroductiontoProfessorG.H.Hardy,alegendaryfigureinmathematics.In Lwówhisdiscoveriesinthetheoryofnumberswerewellknown,andmyfriendSchreierusedtopresenthispapersinseminars.StoriesaboutHardy'seccentricities werewidelytold. IfoundthatbelongingtotheuppermiddleclassdidoftenfacilitatethingsinEngland.InDover,when,bymistake,Ilefttheboatthroughthewrongdoor,twoBritish plainsclothesmeninterceptedmeandwantedtoknowwhereIwasgoing.Imusthavelookedyoungerthanmytwenty­fiveyearsforoneofthemaskedmewhatmy father'soccupationwas.WhenIrepliedthathewasabarrister,themanturnedtohispartnerandsaidinatypicallyBritishway:"Heisallright,hisfatherisabarrister." Ithoughtitwasverycomicalthattheytookmywordsoeasilyforthispieceofinformation. AfterafewhoursinLondon,ItookaneveningtrainforCambridge.Thetrainstoppedeveryfewminutesatstations,allinthedark,whosenameswerenotvisible.I askedayoungmaninmycompartment:"Howcanyoutellwhenit

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isCambridge?''Hethoughtforamomentandreplied:"Iamafraidyoucan't."Afteranothersilence,Itriedtostartanewconversationbyaskinghimwhathethought aboutthepoliticalsituationandwhetherhethoughtEnglandwouldinterveneintheRuhrandhelpFrance.Heponderedagainforaminuteortwoandanswered:"Iam afraidnot!"Iwasabsolutelydelightedbywhatseemedtomesuchvery,veryBritishutterances.AsmyknowledgeofBritishmoresderivedmainlyfromDorothy SayersandAgathaChristienovels,somehowthisfittedin. IgotoffatCambridgeandwenttoahotelcalledtheGardenHousewhichhadbeenrecommendedbyGrossmaninZürich.Sincemyfatherwasfinancingmytravels, eachweekIreceivedfiveortenpoundsatBarclay'sbankfrommyuncle'sbankinLwów.Inthosedaysthiswasalmostaffluence.IwalkedaroundCambridge, admiringtheUniversitybuildingsandlookingintobookstores.(Ialreadyhadapronouncedbook­buying—or,atleast,book­handling—mania.)TheSherlockHolmes andConanDoyleatmosphereIsawinmanyplacesenchantedme. Ihuntedupafewmathematicians.Besicovitch,aRussianémigréfromtheRussianRevolution,wasonewithwhomIhadcorresponded.Hehadsolvedoneofmy problemswhichhadappearedinFundamentaandhadpublishedapaperonit.Itwasreallythefirstnon­obviousexampleofan"ergodictransformation,"amapping ofaplaneontoitself,inwhichthesuccessiveimagesofapointweredenseinthewholeplane. BesicovitchinvitedmetovisithiminhisroomsinTrinityCollege.WhenIenteredhisplace,hesaidnonchalantly,"Newtonlivedhere,youknow."Thisgavemesucha shockthatIalmostfainted.LandmarksinthegreathistoryofsciencelikethisliterallykeptmeinastateofexcitementfortherestofmystayinEngland. BesicovitchandItalkedmathematics.Iwonderifmanyolderpersonswereaccustomedtosuchyoungmencomingintotheirroomsandabruptlyplunginginto scientificprob­

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lemsandtheoremswithoutevenexplainingtheirownpresenceorexchanginggreetingsfirst.MyfriendErdösisstilllikethatattheageofsixty.VonNeumanntoo, whowassourbaneandinterestedinpoliticsandgossip,wouldoftenshiftabruptlyfromageneralconversationtotechnicalscientificremarks. Inseveralways,mystayinCambridgewasoneofthemostpleasantperiodsofmylife—intellectuallyandinapsychologicalsense.Besicovitchinvitedmetoadinner atHighTableatTrinityCollege.Thisdinnerwasoneofthehighpointsofmyentirelifeuntilthen.PresentwereG.H.Hardy,J.J.Thomson,ArthurS.Eddington,and otherfamousscientists,andthereIwas,sittingonlyafewfeetaway.Theconversationwasexciting.Ilistenedtoeveryword.WesatunderanoldportraitofHenry VIII.Foodwasservedinancientsilverdishes.InoticedthatBesicovitchatewithanexcellentappetite.Afterdinnerwemovedtoanotherroom,andhedrankbrandy afterbrandy,whiletheotherscastfurtivebutadmiringglancesinhisdirection. Hardytoldanecdotes,oneofwhichIremember.Asayouthhewasoncewalkingthroughathickfogwithamanoftheclothandtheysawaboywithastringanda stick.Hardy'sclergymancomparedthistotheinvisiblepresenceofGodwhichcanbefeltbutnotseen."Yousee,youcannotseethekiteflying,butyoufeelthepull onthestring."Hardyknew,however,thatinafogthereisnowindandsokitescannotfly.Hardybelievedthat,inmathematics,theCambridgeexaminationscalled "triposes"werenonsensical.Asademonstration,hepersuadedGeorgePolya(who,ifanything,wasamasterofcomputationandmanipulationinclassicalanalysis)to takethemathematicstriposwithoutpreviouscoaching.Polyasupposedlyfailedmiserably. ImetSubrahmanyanChandrasekhar,abrilliantyoungastrophysicistfromIndia.WehadafewmealstogetheratTrinity,wherehewasafellow.Hecollaboratedwith Eddingtonforwhomhehadmixedfeelingsofadmirationandrivalry.Ayearlater,thevacancyintheSocietyofFellowsat

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HarvardwhichIwasinvitedtofillresultedfromChandrasekhar'sacceptanceofanassistantprofessorshipinChicago. WemetagainmuchlaterwhenhewasaconsultantinLosAlamosworkingonthetheoryofturbulenceandotherhydrodynamicalproblems.Chandra,asheisknown amonghisfriends,isoneoftheworld'smostbrilliantandprolificmathematicalastronomers.Hisbooksareclassicsinhisfield. DuringthisstayinCambridge,Michaelmasterm1934,theuniversityortheauthoritiesoftheindividualcollegesforwomen—GirtonandNewnham—abolishedtheold rulewhichforbademenlecturersonthecollegepremises.Iwasinvitedtogiveaseminarontopology.Iwas,ifIamnotmistaken,thefirstmaleinthehistoryofGirton tocrossitsthresholdtogivealecture. OfallthescientistsIhadknowninPoland,theonlyoneIsawwhileinCambridgewasLeopoldInfeld,whowasadocentinLwów.Iknewhimfromourcoffee houses,andwesaweachotherafewtimesinCambridge. Infeldwastall,welloversixfeet,quiteportly,withalargeheadandalargeface.HewasJewish,fromasimpleorthodoxbackground.Inhisautobiographyhe devotedmuchspacetoadescriptionofhisfighttoachieveaneducationandanacademicposition,neitherofwhichwaseasilyattained. Hewasrathergayandwitty.Irememberwhatseemedtomeabrightremarkhemadeafteramonth'sstayinEnglandaboutthedifferencebetweenPolishandEnglish "intellectual"conversations.HesaidthatinPolandpeopletalkedfoolishlyaboutimportantthings,andinEnglandintelligentlyaboutfoolishortrivialthings. Infeldwasaveryambitiousmanandhadacolorfulcareer.Idonotthinkhistalentforphysicsormathematicswasquiteuptohisambitions.InPoland,Ihadhad somedoubtsabouthisrealunderstandingofthemathematicsofthedeeperpartsofgeneralrelativity.Perhapsitwasbecauseofhisratherlimitedbackgroundin fundamentalmathematics.

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ThepopulararticleshewroteinoneoftheWarsawnewspaperswerewellwrittenbut,itseemedtome,notalwaysmathematicallyexact.Atthetimemysightswere setveryhighandIexpectedevennewspaperarticlesaboutsciencetobecomparabletoPoincaré'swonderfulwritingsonpopularscienceorEddington'sexplanations ofrelativitytheoryforpopularaudiences. InfeldcametoPrincetonlater,afewweeksafterIdid,andcollaboratedwithEinsteinonthewell­knownEinstein­Infeldbookonphysics,whichbecameabestseller. HehadmetEinsteininBerlin,andinhisautobiographyhedescribeshowimpressedhehadbeenbyhisfriendlinessandabilitytoputpeopleatease.InPrincetonI hardlysawhim;hewasnotpartofthevonNeumanncrowd. TheCambridgearchitecture,themedievalbuildings,thebeautifulcourtyards,thewalksItookthroughthetown,somewithL.C.Young(nowaprofessoratthe UniversityofWisconsin),arestillamongthestrongestvisualimpressionsofmylife.LikemywalksthroughtheParisoftheFrenchRevolution,thesehavesomehow influencedmytastes,associations,readings,andstudiestothisday. Earlyin1935,IreturnedfromCambridgetoPoland.Itwasnowtimetothinkseriouslyaboutauniversitycareer,althoughthosewere,difficulttimesinwhichtofind evenamodest"docent"position.Aseriesofaccidentalletterswastochangethis;inoneofthem,luckilyforme,IreceivedaninvitationtovisittheUnitedStates.

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PARTII AWORKINGMATHEMATICIANINAMERICA

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Chapter4 PrincetonDays 1935–1936 IfirstheardaboutJohnvonNeumannfrommyhighschoolteacherZawirski.KuratowskialsodescribedvonNeumann'sresultsandhispersonality.Hetoldmehow inaBerlintaxicabvonNeumannhadexplainedinafewsentencesmuchmorethanhe,Kuratowski,wouldhavegottenbycorrespondenceorconversationwithother mathematiciansaboutquestionsofsettheory,measuretheory,andrealvariables.Banach,too,talkedabouthim.Hetoldmehowatthe1927Lwówmeetingheand othermathematicians,Stozekamongthem,hadmadevonNeumanndrunkatthecongressbanquetbyplyinghimwithvodka,totheextentthathehadtoleavethe tabletogotothetoilet.Hecamebackandcontinuedthemathematicalconversationwithouthavinglosthistrainofthought. Itwasonlytowardtheendof1934thatIenteredintocorrespondencewithvonNeumann.HewasthenintheUnitedStates,averyyoungprofessorattheInstitute forAd­

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vancedStudiesinPrinceton.Iwrotehimaboutsomeproblemsinmeasuretheory.HehadheardaboutmefromBochner,andinhisreplyheinvitedmetocometo Princetonforafewmonths,sayingthattheInstitutecouldoffermea$300stipend.ImethimshortlyaftermyreturnfromEngland. Inthefallof1935atopologyconferencehadbeenorganizedinMoscow.Alexandroffinvitedmetoattend.AtthattimerelationsbetweenPolandandSovietRussia werestrained.PassportapplicationsinPolandfortraveltoRussiainvolvedsomuchredtapethatIdidnotreceivemypassportintimeandthusmissedgoingtothe meeting.VonNeumannwrotemethathewouldbepassingthroughWarsawonhiswaybackfromMoscow,suggestingthatwemeetthere.SamuelEilenberg,a youngWarsawmathematicianwellknownforhisveryingenioustopologicalresults,andIwenttomeetthereturningwesterngroup.AtthestationvonNeumann (whomIwasseeingforthefirsttime)wasaccompaniedbytwoAmericanmathematicians,GarrettBirkhoffandMarshallStone.WeallconversedinEnglish. Eilenbergspokeitalittle;IspokeitadequatelythankstomyCambridgestay.VonNeumannwouldbreakintoGermanoccasionally. FromKuratowski'sdescription,Ihadimaginedhimtobeslim,asheapparentlyhadbeenin1927.Hewasinsteadratherplump,thoughnotascorpulentashewasto becomelater.Thefirstthingthatstruckmeabouthimwerehiseyes—brown,large,vivacious,andfullofexpression.Hisheadwasimpressivelylarge.Hehadasort ofwaddlingwalk.(ThisremindsmethatwhenIfirstsawhisgrandson,Malcolm,thesonofhisdaughterMarina,Ifoundituncannytowatchthislittlethree­year­old perambulatedownalonghotelcorridorwithhisgrandfather'swaddle,holdinghishandsbehindhisbackexactlylikeJohnny.Sincehehadbeenbornafterhis grandfather'sdeath,hecouldnotpossiblyhavebeenimitatinghim.Itwouldseemthatgestures,motions,andothertime­dependentphenomena—notmerely

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staticcharacteristicsormattersofspatialconfiguration—canbetransmittedgenetically.) VonNeumannappearedquiteyoungtome,althoughhewasinhisearlythirties,somefiveorsixyearsolderthanI.(Ihavealwayshadmixedfeelingstowardpeople olderthanmyself:ontheonehand,somethinglikerespect;ontheother,aslightfeelingofsuperiority,ofhavingagreatershareinthefuture.)AtonceIfoundhim congenial.Hishabitofinterminglingfunnyremarks,jokes,andparadoxicalanecdotesorobservationsofpeopleintohisconversation,madehimfarfromremoteor forbidding. Duringthisbriefvisit,Stone,vonNeumann,andBirk­hoffgaveajointseminarattheWarsawsectionofthePolishMathematicalSociety.Theirsubjectwasthelattice theoreticalfoundationsofquantumtheorylogic.VonNeumanngavemostofthelecture,Birkhofftalkedbriefly,andStoneaskedquestions.ActuallyIhadmixed impressionsaboutthistalk.Iwasnotatallconvincedthatithadmuchtodowithnovelphysicalideas.Infact,Ithoughtthepointswereabitstretchedandthebig notionofquantumtheorylogicabitartificiallycontrived.IhadanumberofotherconversationswithJohnny,mainlyonmeasuretheory(aboutwhichIhadsenthim reprintsofmyearlierpapers).WealsotalkedalittleabouthisrecentworkonthetheoryofHilbertspaceoperators,althoughIwasnotespeciallyknowledgeable aboutorinterestedinit.ThenhegavemesomepracticaladviceaboutmyforthcomingtriptoPrinceton. InconnectionwiththeMoscowtopologicalmeeting,severalyearsafterWorldWarII,IreceivedaletterfromtheFrenchmathematicianLeray,whowiththeLwów mathematicianJuliuszSchauderhadwrittenacelebratedpaperonfixedpointsfortransformationsinfunctionspacesandapplicationsinthetheoryofdifferential equations.Schauder,ourmutualfriend,wasmurderedbytheNazis.LeraywantedtohaveaphotographofhimforhimselfandforSchauder'sdaughterwhosurvived thewarandlivesinItaly.Buthe

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couldnotfindanyinPolandoranywhereandhewrotemeaskingwhetherImighthaveasnapshot.SomemonthsafterJohnnyvonNeumann'sdeathIwaslookingat someofthebooksinhislibraryandagroupphotooftheparticipantsintheMoscowconferencefellout.Schauderwasthere,aswereAlexandroff,Lefschetz, Borsuk,andsomedozenothertopologists.IsentthisphotographtoLeray.Ithassincebeenreproducedinseveralpublications. JustasinLwów,theWarsawmathematiciansgatheredinapastryshopanddiscussedmathematicsforhours.TheyalsofrequentedthefamousFukerwineshopinthe oldtown.ThisiswhereEilenbergandItookJohnnyandhiscompanionstodrinkFuker'scelebratedhydromel.Hereheentertaineduswiththestoryofhowatthe requestofPrincetonfriendshehadboughtseveralpoundsofcaviarinMoscowtobringbacktotheU.S.andhadaskedastewardtostoreitintheiceboxofthe restaurantcar.Inthemorningwhentheywokeup,inPoland,theydiscoveredthattherestaurantcarhadbeenuncoupledatthePolish­Russianborder.Theywere returningtotheStatescaviarless!HetalkedalsoabouthisowndecisiontoemigratetoAmericaandthegeneralimpracticalityandlackofforesightofEuropean scientists.IntheGermanuniversitiesthenumberofexistingandprospectivevacanciesforprofessorshipswasextremelysmall—somethingliketwoorthreeinthe entirecountryforthenexttwoyears.Yetmostofthetwoorthreescoredocentscountedonobtainingaprofessorshipinthenearfuture.Withhistypicalrational approach,vonNeumanncomputedthattheexpectednumberofprofessorialappointmentswithinthreeyearswasthree,whereasthenumberofdocentswasforty. Thisiswhathadmadehimdecidetoemigrate,nottomentiontheworseningpoliticalsituation,whichmadehimfeelthatunhamperedintellectualpursuitswould becomedifficult.In1930,heacceptedanofferofavisitingprofessorshipatPrincetonUniversity,and,in1933,shortlyafterthecreationoftheInstitute,he

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wasinvitedtobecometheyoungestmemberofthepermanentfacultyoftheInstituteforAdvancedStudies. InDecember1935IsailedontheEnglishshipAquitaniafromLeHavreonmyfirsttransatlanticcrossing.Theweatherwasbeautifulforthefirsttwodays;thena violentstormsprangup,andIbecameseasick.WhentheboatapproachedNewYork,theseacalmeddownandmysea­sicknessstopped. AftertwodaysinNewYork,ItriedtoreachvonNeu­manninPrincetonbutgotnoanswer,soIcalledtheInstitute.Itwasquiteanexperienceformetogointoan Americantelephoneboothforthefirsttime.Whentheoperatorsaid,"Holdthewire!"Ididnotunderstandtheexpressionandasked"WhichwireshouldIhold?"I reachedSolomonLefschetz,aprofessorattheUniversity,whotoldmehowtogettoPrincetonfromNewYork.Hesaiditwasveryeasy,thatthereweretrains everyhour.Icouldnotunderstandthat.PrincetonIknewwasaverysmalltown,whyshouldtherebeatraineveryhour?Ididnotknowitwasonthemainlineto PhiladelphiaandWashington. InPrincetonIwentstraighttoregisterattheInstitutewhichwashousedinFineHall,aUniversitybuilding,foritdidnothaveitsownquartersyet.Ayoungandpretty MissFlemmingandanolderMissBlakereceivedme.Iwasgreetedwithsmiles.Itsurprisedme,andIwonderediftherewassomethingalittlefunnyaboutthewayI wasdressedorwhethermytrouserswerenotproperlybuttoned(therewerenozippersinthosedays). IcheckedinataboardinghouseandwentdirectlytovisitvonNeumanninhislargeandimpressivehouse.Ablackservantletmein,andtherewasSolomonBochner intheliving­roomandababycrawlingonthefloor.(ThebabywasvonNeumann'sdaughterMarina,sixmonthsoldatthetime.)Marietta,hisfirstwife,alsoa Hungarian,greetedme.IknewofBochnersincewehadcorrespondedaboutmathematics.BochnerandvonNeumannweretalkingpolitics.

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VonNeumannexpressedgreatpessimismaboutthepossibilityofawarinEurope.(Thiswasaboutthreeyearsbeforetheactualoutbreak.)Apparentlyhehada ratherclearpictureofthecatastrophestocome.HesawRussiaasthechiefantagonisttoNaziGermany.BelievingthattheFrencharmywasstrong,Iasked,''What aboutFrance?""Oh!Francewon'tmatter,"hereplied.Itwasreallyveryprophetic. MylodgingswereinaboardinghouseonVandevanterStreet,ifIremembercorrectly.Thereweresixoreightothermenthere,notallstudents,andweatetogether.I rememberhowtheconversationwasatfirstcompletelyincomprehensibletomealthoughIknewEnglish.TheAmericanaccenttookmebysurprise,andImissed mostofwhatwasbeingsaid.ThenafteraweekIunderstoodeverything.Thisisacommonexperience,notonlywithlanguagesbutalsowithmathematics—a discontinuousprocess.Nothing,nothing,atfirst,andsuddenlyonegetsthehangofit. IbecameafrequentvisitoratthevonNeumannswhowereverysociableandheldpartiestwoorthreetimesaweek.Thesewerenotcompletelycarefree;theshadow ofcomingworldeventspervadedthesocialatmosphere.ThereImettheAlexanders,whoweregreatfriendsofthevonNeumanns.JamesAlexander,alsoa professorattheInstitute,wasanoriginaltopologist,thecreatorofnovelproblemsandstrange"pathological"examplesoftopologicalobjects.Hewasthescionofa wealthyfamilyandveryeccentric. Atapartyafewweekslater,Isawamanwhomusthavebeenfiftybutseemedtomeinfinitelyold—Iwastwenty­sixatthetime.Hewassittinginabigchairwitha niceyoungladyonhisknee.Theyweredrinkingchampagne.IpassedbyJohnnyandasked,"Whoisthisgentleman?""Oh!don'tyouknow?HeisvonKárman,the famousaerodynamicist."VonKármanwasoneofJohnny'sfriends.Andheadded:"Don'tyouknowthatheinventedconsulting?"VonKármanwasoneofthefirst scientistswholearnedtoflyaplaneinthefirstWorldWar.Hetoldmethathehadoneofthelow

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numberedinternationalflyinglicenses.Hisflyingexperiencesdirectlyinfluencedhisideasaboutjetengines,whichbecamesoimportantinWorldWarIIdevelopments. MuchlaterIcametoknowhimquitewell.Heusedtosaythatengineersarepeoplewhoperpetuatethemistakesmadeinthepreviousgeneration.In1968Ifound myselfwithhiminIsraelatameetingonhydrodynamics.Bythenhewasaricholdman.Itwashisfirstvisittothatcountry,andhewassomovedandimpressedby whathesawthathegaveawayfive­andten­dollarbillsastipstowaitersandtaxidrivers,nomatterwhatthesizeofthecheck. Johnnywasalwaysimpressedbypeoplewhoweresuccessfulinpoliticalororganizationalactivitiesorinphysicalexploits,andliecultivatedthem.ItwasJohnnywho saidaswewalkedpasttheelegantGothicPrincetonUniversityChapel:"Thisisourone­million­dollarprotestagainstmaterialism."Idonotknowwhetherthisbon motwashisownornot,butitshowshissenseofhumoraboutmoney. Inthosedayshestillcalledme"Mr.Ulam."Oncehesaidtomeasweweredrivingintherainandwerecaughtinatrafficjam,"Mr.Ulam,carsarenogoodfor transportationanymore,buttheymakemarvelousumbrellas."IoftenrememberthiswhenIamcaughtinthetrafficjamsoftoday.Johnnyalwayslovedcarsbuthe drovesomewhatcarelessly. Johnnylivedrathersumptuously.TheprofessorsattheInstitutewerethehighestpaidacademicsintheUnitedStates—paidevenmorethanatHarvard.Thistended tocreateanimositiesbetweenInstituteandUniversityprofessors.Alsotheircompensationwasinsharpcontrasttothealmostnegligiblestipendsofferedtothefellows andvisitorsoftheInstitute. Thegreatname,thegreatcelebrity,thegreatlight,sofarasthegeneralpublicwasconcerned,was,ofcourse,AlbertEinstein.Ifirstmethisassistant,Mayer,a mathematicianandastrangeperson.ThenIwasintroducedtoEinsteinhimselfandnoticedhisratherpeculiarEnglish.Hewould

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say:"Heisaverygoodformula,"pointingatsomethingontheblackboard! Acousinofmine,AndrzejUlam,abanker,cametoNewYorkonbusinessabouttwomonthsaftermyarrival,andIinvitedhimtovisitmeinPrinceton.Ithappened thatduringthatweekIwasgivingatalkinsomeseminar,andmynamewaslistedonthesamepageoftheInstitute'sBulletinastheannouncementofEinstein'sregular weeklyseminar.Thisimpressedhimenormously;hementioneditinaletterhome,andmyreputationamongfriendsandfamilyinPolandwasmade. HermannWeylwasalsoaprofessorattheInstitute.ImethiminPrincetonandwenttohishomeseveraltimes.Hewasalegendaryfigure,mucholderthanvon Neumann.Hehadthesamewidebreadthofinterestswhichimpressesmesomuch.MyfriendGian­CarloRota,nowaprofessoratMIT,toldmemuchlaterthathe hadheardWeyl'soriginalsymmetrylecturesandwasenormouslyimpressed.Theywereabitheavy,butatthesametimetheygaveafeelingofuniversalculture.More recentlyWeyl'spurelymathematicalschemataoralgebraicentitiesfoundessentialapplicationsasmodelsforthepropertiesofthemysteriousneutrinoparticlesandso­ calledweakinteractionsimportantinthebetadecayofnuclei. AfterWeyl'sfirstwifediedheremarriedandlivedinSwitzerlandforatime.Hewasunawareoftherulesandregulationsgoverningthelengthoftimeanaturalized AmericancitizenmayliveabroadwithoutreturningtothiscountryandstillretainhisAmericancitizenship.Helosthisbynegligence.Whenithappened,everyonewas shocked.MembersoftheMathematicalSocietyandtheNationalAcademyofScienceswantedtohavehimreinstatedasaU.S.citizen.Thisrequiredaspecialbillin Congress,andsomefriendsaskedmetointervenewithSenatorAnderson,whomIknewwell,tohelpinthismatter.InthemeantimeWeylhadcollapsedinastreetin Zürichwhileputtingaletterinamailboxanddiedofaheartattack.

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EverydayIwenttotheInstituteforfiveorsixhours.BythattimeIhadquiteanumberofpublishedpapers,andpeoplethereknewsomeofthem.Italkedwith Bochneragooddealandsoonaftermyarrivalcommunicatedtohimaproblemabout"InvertingtheBernouillilawoflargenumbers."Bochnerprovedthetheoremand publisheditinTheAnnalsofMathematics.(Bytheway,thisproblemisstillsolvedinasimplecaseonly.Theinverseofthelawoflargenumbers,thoserequiringa measureinaspaceofmeasures,hasnotyetbeenproved.) Iwenttolecturesandseminars,heardMorse,Veblen,Alexander,Einstein,andothers,butwassurprisedhowlittlepeopletalkedtoeachothercomparedtothe endlesshoursinthecoffeehousesinLwów.Therethemathematiciansweregenuinelyinterestedineachother'swork,theyunderstoodoneanotherbecausetheir workrevolvedaroundthecentralthemeofsettheoreticalmathematics.Here,incontrast,severalsmallgroupswereworkinginseparateareas,andIwassomewhat disappointedatthislackofcuriosityeventhoughtheInstituteandtheUniversityhadaveritablegalaxyofcelebrities,possiblyconstitutingoneofthegreatest concentrationsofbrainsinmathematicsandphysicsevertobeassembled.Beingamaliciousyoungman,ItoldJohnnythatthisremindedmeofthedivisionofrackets amongChicagogangsters.The"topologyracket"wasprobablyworthfivemilliondollars;the"calculusofvariationsracket,"motherfive.Johnnylaughedandadded, "No!Thatisworthonlyonemillion." TherewasanotherwayinwhichthePrincetonatmospherewasentirelydifferentfromwhatIexpected:itwasfastbecomingawaystationfordisplacedEuropean scientists.Inaddition,thesewerestilldepressiondaysandthesituationinuniversitiesingeneralandinmathematicsinparticularwasverybad.Peoplewithimpressive backgroundsandgoodcredentials(notonlyvisitorslikeme,butnative­bornAmericans)werestillwithoutjobsseveralyearsaftergettingtheirdoctorates.Averyable mathematicianand

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logicianfriendofmine,whoisnowamemberoftheNationalAcademyofSciences,wasthenonamiserablestipendatPrincetonwaitingforapositiontoopen somewhere.Onedayatelegramcameofferinghimaninstructorshipattwelvehundreddollarsayear.Hetoldmehethoughthewasdreamingandquicklyaccepted thejob.Thereweremanysuchcases.AtthattimeIwastoldthatthreepersons"owned"theAmericanMathematicalSociety:OswaldVeblen,G.D.Birkhoff,and ArthurB.Coble,fromIllinois.Mostacademicpositionsweresecuredthroughtherecommendationsofthesethree.Whatacontrasttothevastnumberofuniversity jobsinmathematicswhichexisttoday! ItwasVeblenwhowasresponsibleforJohnny'spresenceattheInstitute.Hehadinvitedhimforasemester'sstayatfirst,andlaterarrangedforhimtoremain.He likedJohnnyverymuchandconsideredhimalmostasason. OswaldVeblen,anephewofThorsteinVeblen,authorofTheTheoryoftheLeisureClass,wasafamousAmericanmathematician,tall,slim,Scandinavian­looking, withacausticsenseofhumor.Hewaswellknownforhisworkonthefoundationsofprojectivegeometryandtopology. VeblenorganizedwalksinthePrincetonwoods,andIwasinvitedtojoinsomeoftheseexpeditionsduringwhichtherewaslotsofmathematicaltalkandgossipwhile hecutdeadwoodandtreebranchestoclearthepaths. TomymindthePrincetonwoodswiththeirthin,spindlytreesandmarsheswerenotatallimpressivecomparedtothePolishforests.ButitwasthefirsttimeIheard andsawwhatseemedlikegiganticKafkaesquefrogs.Thebirdswerealsoverydifferent,andIfelttrulyonanothercontinent,inaveryexoticland. Throughoutthesewalksanddiscussionstherealwayslurkedinthebackofmymindaquestion:wouldIreceiveaninvitationfromsomeAmericaninstitution,which wouldenablemetoremain?Moresubconsciouslythanconsciously,Iwaseagerlylookingforawaytostay,thereasonbeingthecriticalpoliticalsituationinEurope andthecatastrophicjob

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situationformathematiciansthere,especiallyJews.TherewasverylittlefutureformeinPolandanditwasbecomingincreasinglyevidentthatthecountrywasinmortal danger.Ialsoadmiredthefreedomofexpression,ofwork,thesenseofinitiative,thespiritwhichwasintheair;herethefutureoftheworldwasbeckoning.Even thoughIdidnotmentionanyofthisexplicitlytoJohnny,Ibecameveryeagertostayandtotakeworkifitpresenteditself. Aboutthattime,KuratowskiappearedinPrincetonforamonth'svisit.Hearrivedlateinthespring,andwonderedwhethertherewasanychancethatImightbe invitedtoremainintheStatesforthenextacademicyear.HewenttoHarvardtogivealecture,andseveraloftheprofessorsthere,Birkhoff,Graustein,andothers, askedhimaboutme.Heprobablygavethebestreferences.Whenhespoketomeaboutthesepossibilities,hehadmixedfeelings.Heknewfullwellthattherewas verylittlechanceofaprofessorshipformeinPoland,andrealizeditwasgoodformyfuturecareertoremainintheStatesawhilelonger,yethewasgenuinelysorryat thethoughtthatImightnotreturn. Duringthisstay,heandJohnnyobtainedsomeverystrongresultsoncertaintypesofprojectivesets.Thisisaveryeleganttheoryofoperationsinmathematicallogic, goingbeyondtheAristotelianorBooleanones.Tothisdaythetheoryisfullofmysterioussituationsveryfundamentaltoproblemsinthefoundationsofmathematics andsettheory.Muchrecentworkconcernssuchprojectiveoperations,andsomerecentresultscertainlyoriginatefromthisinterestingpaper.Itiscurioushowthis cameabout.Withhistechnicalvirtuosityanddepthofpenetration,onceJohnnyhadreceivedthestartingimpulsehewasabletofindthedecisivepoints.Agood exampleofhowcollaborationinmathematicsisveryoftenfruitful! VonNeumanninvitedmeonedaytogiveatalkinhisseminaronmyresultsin"semi­simplegroups,"whichwasasubjectIdidnotknowverymuchabout.Ihave oftensucceededinobtainingratheroriginalandnotunimportantre­

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sultsinareaswhereIdidnotknowthefoundationsorthedetailsofatheorytoowell.AtthisseminarJohnnyaskedmesomeverysearchingandpenetratingquestions, andIhadtothinkveryhardtogivesatisfactoryanswers;Ididnotfeelhewasdoingthistoembarrassme,butonlybecauseofhisoverridingobjectivityanddesireto makethingsclear. AtsomelecturesvonNeumannsometimes"snowed"thestudentsbyelaboratingtheeasierpointsandquicklyglossingoverthedifficulties,buthealwaysdemonstrated hisfantasticandtosomeextentpropheticrangeofinterestsinmathematicsanditsapplicationsandatthesametimeanobjectivitywhichIadmiredenormously. Asamathematician,vonNeumannwasquick,brilliant,efficient,andenormouslybroadinscientificinterestsbeyondmathematicsitself.Heknewhistechnicalabilities; hisvirtuosityinfollowingcomplicatedreasoningandhisinsightsweresupreme;yethelackedabsoluteself­confidence.Perhapshefeltthathedidnothavethepower todivinenewtruthsintuitivelyatthehighestlevelsorthegiftforaseeminglyirrationalperceptionofproofsorformulationofnewtheorems.Itisveryhardformeto understandthis.Perhapsitwasbecauseonacoupleofoccasionshehadbeenanticipated,preceded,orevensurpassedbyothers.Forinstance,hewasdisappointed thathehadnotfirstdiscoveredGödel'sundecidabilitytheorems.Hewasmorethancapableofthis,hadheadmittedtohimselfthepossibilitythatHilbertwaswrongin hisprogram.Butitwouldhavemeantgoingagainsttheprevailingthinkingofthetime.AnotherexampleiswhenG.D.Birkhoffprovedtheergodictheorem.Hisproof wasstronger,moreinteresting,andmoreself­containedthanJohnny's. DuringmystayinPrincetonIfeltthattherewassomehesitationonJohnny'spartabouthisownwork.Hewasimmersedinhisnewworkoncontinuousgeometries andinthetheoryofclassesofoperatorsinHilbertspaces.ImyselfwasnotsointerestedinproblemsconcerningpropertiesofHilbertspaces.Johnny,Icouldfeel, wasnotcompletelycer­

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tainoftheimportanceofthiswork,either.Onlywhenhefound,fromtimetotime,someingenious,technicallyeleganttrickoranewapproachdidheseemvisibly stimulatedorrelievedofhisowninternaldoubts. Itwasatthattimethathebegantothinkofproblemsawayfrompuremathematics,althoughthiswasnotthefirsttimeinhislife.(Hehadwrittenhisfamousbookon themathematicalfoundationsofquantumtheoryin1929.)Hewasthinkingnowmoreaboutclassicalproblemsinphysics.Forexample,hestudiedproblemsof turbulenceinhydrodynamics.Inhiscontinuousgeometries,theirelementsdonotplaytheroleofwhatwenormallyconsideras"points"inEuclideanspace;itisa creationofa"pointless"geometry,anamewhichlentitselftomanyaneasyjoke. Hecamebackagainandagaintothepossibilitiesofreformulatingthelogicofquantumtheory,thesubstanceofthetalkhegaveattheWarsawseminar.InPrinceton hefrequentlyworkedonthistopic.Listeningtohisconversation,Isaworfelthisownhesitation,andIsufferedfromdoubt,too,becausetherewasnodefinite experimentalpossibilityofverifyingthis—itseemedpurelyaquestionoflogic.Purely"grammatical"approachesneverinterestedmemuch.Whensomethingismerely convenientortypographicallyuseful,itseemslessinterestingtomethanwhenthereisamorerealphysicalbase,orifabstract,stillsomehowpalpable.Ihavetoadmit thattherearecaseswhereformalismbyitselfhasgreatvalue—forexample,thetechnique,orratherthenotationofFeynmangraphsinphysics.Itisapurely typographicalidea,itdoesnotbringinitselfanytangibleinputintoaphysicalpicture,nevertheless,bybeingagoodnotationitcanpushthoughtsindirectionsthatmay proveusefulorevennovelanddecisive.Beyondthis(andextremelyimportant),thereisthemagicof"algorithms,"orsymbolisminmathematics.Calculusitselfshows thewonderofit.Varioustransforms,generatingfunctions,andthelikeperformalmostmiraculouslyinmathematicalapplications.

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VonNeumannwasthemasterof,butalsoalittlebittheslaveto,hisowntechnique.Whenhesawthatsomethingcouldbedone,helethimselfbecarriedawayon tangents.Myownfeelingisthatsomeofhismathematicalworkonclassesofoperatorsoronquasi­periodicfunctions,forexample,isveryinterestingtechnically,but tomytastenotterriblyimportant;hecouldnotresistdoingitbecauseofhisfacility. Howterriblyimportanthabitis.Itmaylargelydeterminethecharacteristicsorthenatureofthebrainitself.Habitsinfluenceorperhapscanlargelydeterminethechoice oftrainsofthoughtinone'swork.Oncetheseareestablished(andinmyopiniontheymaybeestablishedveryquickly—sometimesafterjustafewtrials),the "connections"or"programs"or"subroutines"becomefixed.VonNeumannhadthishabitofconsideringthelineofleastresistance.Ofcourse,withhispowerfulbrain hecouldquicklyvanquishallsmallobstaclesordifficultiesandthengoon.Butifthedifficultywasgreatrightfromthestart,hewouldnotknockhisheadagainstthe wall,norwouldhe—asIonceexpressedittoSchreier—walkaroundthefortressandknockhereandtheretofindtheweakestspotsandtrytobreakthrough.He wouldswitchtoanotherproblem.OnthewholeinhisworkhabitsIwouldcallJohnnymorerealisticthanoptimistic. Johnnywasalwaysahardworker;hehadagreatenergyandtoughnessbehindaphysicalappearancethatwassomewhatonthesoftside.Eachdayhewouldstart writingbeforebreakfast.Evenatpartiesinhishouse,hewouldoccasionallyleavethegueststogotohisstudyforhalfanhourorsotorecordsomethingthatwason hismind. Hemaynothavebeenaneasypersontolivewith—inthesensethathedidnotdevoteenoughtimetoordinaryfamilyaffairs. Somepeople,especiallywomen,foundhimlackingincuriosityaboutsubjectiveorpersonalfeelingsandperhapsdeficientinemotionaldevelopment.Butinhis conversationswithme,Ifeltthatonlyacertainshynessprevented

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himfromhavingmoreexplicitdiscussionsalongtheselines.Suchseemingdiffidenceisnotuncommonamongmathematicians.Non­mathematiciansoftenreproachus forthisandmayresentthisapparentemotionalinsensitivityandexcessivequantitativeandrationalbent,especiallyinattitudestowardsmundanemattersoutside science.VonNeumannwassobusywithmathematics,physics,andwithacademicaffairs,nottomentionincreasinglyinnumerableactivitieslateronasaconsultantto manyprojectsandGovernmentadvisorywork,heprobablycouldnotbeaveryattentive,"normal"husband.Thismightaccountinpartforhisnot­too­smoothhome life. Tobesure,hewasinterestedinwomen,outwardly,inapeculiarway.Hewouldalwayslookatlegsandthefigureofawoman.Wheneveraskirtpassedbyhewould turnandstare—somuchsothatitwasnoticedbyeveryone.Yetthiswasabsentmindedlymechanicalandalmostautomatic.Aboutwomeningeneralheoncesaidto me,"Theydon'tdoanythingverymuch."Hemeant,ofcourse,nothingmuchofimportanceoutsideoftheirbiologicalandphysiologicalactivities. HedidnotshowsocialprejudiceandneverconcealedhisJewishorigins(eventhoughIthinkhehadactuallybeenbaptizedaChristianinhischildhood).Infact,he wasveryproudofthebirthofthestateofIsraelin1948andwaspleasedbytheJewishvictoriesoverthesurroundingArabcountries—asortofmisplaced nationalism. Hisfather,abanker,hadbeentitled"von."IntheAustro­HungarianEmpirepeoplewererewardedwithtitlesbutthesecouldalsobeobtainedbygiftsofmoneytothe government.Johnnyneverusedthefulltitle(neitherdidvonKármanwhowasalsoofJewishorigin).Hewasillateasewithpeoplewhowereself­madeorcamefrom modestbackgrounds.Hefeltmostcomfortablewiththird­orfourth­generationwealthyJews.Withsomeonelikeme,hewouldthenoftenuseJewishexpressionsor jokesasaspicetoconversation.Hewasamanoftheworld,notexactlysnobbish

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butquiteconsciousofhisposition,whofeltmoreateasewithpeoplewiththesamebackground. Hewasbroadlyeducatedandwellversedinhistory,especiallyoftheRomanEmpire—itspowerandorganizationfascinatedhim.Perhapspartofthisinterest stemmedfromamathematician'sappreciationofthedifferencebetweenvariablesinvolvingindividualpoints,orpersons,andgroupsofsuch,orclassesofthings.He wasgiventofindinganalogiesbetweenpoliticalproblemsofthepresentandofthepast.Sometimes,theanalogywasgenuinelythere,butthereweresomanyother differentfactorsthatIdon'tthinkhisconclusionswerealwaysjustified. Ingeneral,hetendednottodisagreewithpeople.Hewouldnotcontradictordissuadewhenaskedforadviceaboutthingstheywereinclinedtodo.Inmattersof ordinaryhumanaffairs,histendencywastogoalong,eventoanticipatewhatpeoplewantedtohear.Healsohadtheinnocentlittletrickofsuggestingthatthingshe wanteddoneoriginatedwiththepersonshewantedtodothem!IstartedusingthisploymyselfafterIlearneditfromhim.However,inscientificmatters,hediddefend theprincipleshebelievedin. Whenitcametootherscientists,thepersonforwhomhehadadeepadmirationwasKurtGödel.Thiswasmingledwithafeelingofdisappointmentatnothaving himselfthoughtof''undecidability."ForyearsGödelwasnotaprofessoratPrinceton,merelyavisitingfellow,Ithinkitwascalled.Apparentlytherewassomeoneon thefacultywhowasagainsthimandmanagedtopreventhispromotiontoaprofessorship.Johnnywouldsaytome,"Howcananyofusbecalledprofessorwhen Gödelisnot?"WhenIaskedhimwhoitwaswhowasunfriendlytoGödel,hewouldnottellme,eventhoughwewereclosefriends.Iadmiredhisdiscretion. AsforGödel,hevaluedJohnnyveryhighlyandwasmuchinterestedinhisviews.IbelieveknowingtheimportanceofhisowndiscoverydidnotpreventGödelfroma

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gnawinguncertaintythatmaybeallhehaddiscoveredwasanotherparadoxàlaBuraliForteorRussell.Butitismuch,muchmore.Itisarevolutionarydiscovery whichchangedboththephilosophicalandthetechnicalaspectsofmathematics. WhenwetalkedaboutEinstein,Johnnywouldexpresstheusualadmirationforhisepochaldiscoverieswhichhadcometohimsoeffortlessly,fortheimprobableluck ofhisformulations,andforhisfourpapersonrelativity,ontheBrownianmotion,andonthephoto­electricquantumeffect.Howimplausibleitisthatthevelocityof lightshouldbethesameemanatingfromamovingobject,whetheritiscomingtowardyouorwhetheritisreceding.Buthisadmirationseemedmixedwithsome reservations,asifhethought,"Well,hereheis,soverygreat,"yetknowinghislimitations.HewassurprisedatEinstein'sattitudeinhisdebateswithNielsBohr—athis qualmsaboutquantumtheoryingeneral.Myownfeelinghasalwaysbeenthatthelastwordhasnotbeensaidandthatanew"superquantumtheory"mightreconcile thedifferentpremises. IonceaskedJohnnywhetherhethoughtthatEinsteinmighthavedevelopedasortofcontemptforotherphysicists,includingeventhebestandmostfamousones— thathehadbeendeifiedandlionizedtoomuch.Noonetriedtogohimonebetterbygeneralizinghistheoryofrelativity,forexample,orinventingsomethingwhich wouldrivalorchangeorimproveit.Johnnyagreed."Ithinkyouareright,"hesaid,"hedoesnotthinktoomuchofothersaspossiblerivalsinthehistoryofphysicsof ourepoch." Comparisonsareinvidious,andthereisnoquestionofanylinearorderofeminenceorgreatnessinscience.Muchofitisaquestionoftaste.Itisprobablyasdifficult tocomparemathematicianslinearlyorotherwiseasitwouldbetocomparemusicians,poets,orwriters.Thereare,ofcourse,largeandobviousdifferencesin"class." Onecouldsafelysay,Ithink,thatHilbertwasprobablyagreatermathematicianthansomeyoungteachingassistantchosenatrandomat

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alargeuniversity.Ifeelthatsomeofthemostpermanent,mostvaluable,mostinterestingworkofvonNeumanncametowardstheendofhislife,involvinghisideason computing,ontheapplicationsofcomputing,andonautomata.Therefore,whenitcomestolastingimpact,Ithinkinmanywaysitmightbeasgreatasthatof Poincaré's,whowas,ofcourse,quitetheoreticalanddidnotactuallycontributedirectlytotechnologyitself.Poincaréwasoneofthegreatfiguresinthehistoryof mathematics.SowasHilbert.Asmathematicians'mathematicians,theyareidolized,perhapsalittlemorethanvonNeumann.Butfinaljudgmentshavetobelefttothe future. OneoftheluckiestaccidentsofmylifehappenedthedayG.D.BirkhoffcametoteaatvonNeumann'shousewhileIwasvisitingthere.Heseemedtohaveheard aboutmefromhissonGarrett,whomIhadmetinWarsaw.Wetalkedand,aftersomediscussionofmathematicalproblems,heturnedtomeandsaid,"Thereisan organizationatHarvardcalledtheSocietyofFellows.Ithasavacancy.Thereisaboutonechanceinfourthatifyouwereinterestedandappliedyoumightreceivethis appointment."Johnnynoddedeagerlyinmydirection,andIsaid,"Yes,ImightbeinterestedinspendingsometimeatHarvard."Amonthlater,inAprilof1936,I receivedaninvitationtogiveatalkatthemathematicscolloquiumthere.ThetalkwasfollowedbyaninvitationtoadinnerattheSocietyofFellows.Isupposethis wastolookmeoverwithoutmybeingawareofit. Atthecolloquium,Italkedaboutsomethingwhichisstillbeingworkedon,theexistenceinmanystructuresofasmallnumberofelementswhichgeneratesubgroups orsubsystemsdenseinthewholestructure.(Or,popularlyspeaking,outofaninfinitevarietyofobjectsonecanpickafewsuchthatbycombiningthemonecan obtain,withonlyasmallerror,alltheothers.)TheresultsweresomethingthatJozefSchreierandIhadprovedacoupleofyearsearlier.Italkedwithconfidence—I don'tremembereverbeingverynervousaboutgivingtalksbecauseIalwaysfeltIknew

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whatIwastalkingabout.Itmusthavebeenwellreceived,forwhenIreturnedtoPrinceton,Ifoundaletterwhichgladdenedmenoend.ItwasfromtheSecretaryof theHarvardCorporation,signedintheEnglishmanner"YourObedientServant."ItwasanominationtothepositionofJuniorFellowtobeginthefollowingautumn andtolastforthreeyears.Theconditionswereextremelyattractive:fifteenhundreddollarsayearplusfreeboardandroomtogetherwithsometravelallowances.In thosedaysitseemedaroyaloffer. Withthisinmypocket,IhappilybeganpreparationstoreturntoPolandforthesummer.TomakeupforJohnny'sdisasterofthesummerbefore,mynewPrinceton acquaintancesgavemeanordertobringbackahugeamountofcaviar.LittledidtheyrealizethatinPoland,whichdidnotproduceit,itwasasexpensiveasinthe West.

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Chapter5 HarvardYears 1936–1939 IcametotheSocietyofFellowsduringitsfirstfewyearsofexistence.GarrettBirkhoffandB.F.Skinner,thepsychologist,wereamongitsoriginalmembers.Mostof theJuniorFellows,aswewerecalled,wereintheirmid­twenties,mainlybuddingpost­doctoralscholars. Iwasgivenatwo­roomsuiteinAdamsHouse,nextdoortoanothernewfellowinmathematicsbythenameofJohnOxtoby.Aboutmyage,hedidnothavehis doctor'sdegreebutwaswellknownattheUniversityofCalifornia—wherehehaddonehisgraduatework—forhisbrillianceandpromise.Itookaninstantlikingto him.Hewasatallish,blue­eyedredhead,withaconstantgooddisposition.Anattackofpolioinhishigh­schoolyearshadseverelycrippledoneleg,sothathehadto walkwithacrutch. HewasinterestedinsomeofthesamemathematicsIwas:insettheoreticaltopology,analysis,andrealfunctiontheory.Rightoff,westartedtodiscussproblems concerning

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theideaof"category"ofsets."Category"isanotioninawayparalleltobutlessquantitativethanthemeasureofsets—thatis,length,area,volume,andtheir generalizations.Wequicklyestablishedsomenewresults,andthefruitsofourconversationsduringthefirstfewmonthsofouracquaintancewerepublishedastwo notesinFundamenta.Wefollowedthiswithanambitiousattackontheproblemoftheexistenceofergodictransformations.Theideasanddefinitionsconnectedwith thishadbeeninitiatedinthenineteenthcenturybyBoltzmann;fiveyearsbeforeworkonthishadculminatedinvonNeumann'spaper,followed(andinaway superseded)byG.D.Birkhoff'smoreimposingresult.Birkhoff,inhistrail­breakingpapersandinhisbookondynamicalsystems,haddefinedthenotionof ''transitivity."OxtobyandIworkedonthecompletiontotheexistenceoflimitsintheergodictheoremitself. Inordertocompletethefoundationoftheideasofstatisticalmechanicsconnectedwiththeergodictheorem,itwasnecessarytoprovetheexistence,andwhatis more,theprevalenceofergodictransformations.G.D.Birkhoffhimselfhadworkedonspecialcasesindynamicalproblems,buttherewerenogeneralresults.We wantedtoshowthatoneverymanifold(aspacerepresentingthepossiblestatesofadynamicalsystem)—thekindusedinstatisticalmechanics—suchergodic behavioristherule. Thenature,intensityandlongdurationofourdailyconversationsremindedmeofthewayworkhadbeendoneinPoland.OxtobyandIusuallysatinmyroom,which wasratherstark,althoughIhadrentedacoupleoforientalrugstofurnishit,orinhisown,whichwasevenmorespartan. Wediscussedvariousapproachestoapossibleconstructionofthesetransformations.Withmyusualoptimism,Iwassomehowsureofourultimatesuccess.Wekept G.D.Birkhoffinformedofthestatusofourattacksontheproblem.HewouldsmilewhenItalkedtohimatdinnerattheSocietyofFellows,partlyamused,partly impressedbyoursingle­mindedpersistence,andpartlyskeptical,thoughhereally

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hadanopenmindaboutourchances.HewouldcheckwhatItoldhimwithOxtoby,amorecautiousperson.Ittookusmorethantwoyearstobreakthroughandto finishalongpaper,whichappearedinTheAnnalsofMathematicsin1941andwhichIconsideroneofthemoreimportantresultsthatIhadapartin. ThechairmanoftheSocietywasL.J.Henderson,afamousbiologist,authorofabook,TheFitnessofEnvironment,whichenjoyedagreatpopularityatthetime, notonlyamongspecialists,butquitegenerally.L.J.,ashewascalled,wasagreatFrancophile.Indeed,theSocietywasmoldedalongthelinesoftheFondationThiers inParis,ratherthanontheCambridgeorOxfordsystemsoffellowsinthecollege. TheSocietywascomposedofsomefiveorsixSeniorFellowsandabouttwenty­twoJuniorFellows. TheSeniorFellowswerewell­knowndistinguishedprofessors,likeJohnLivingstonLowesinliterature,SamuelEliotMorison,thehistorian,Henderson,andAlfred NorthWhitehead,thefamousEnglishphilosopher,whohadalreadyretiredfromhisprofessorshipatHarvardwhenIenteredtheSociety.Ioftenhadthepleasureof sittingnexttohimatthetraditionalMonday­nightdinnersoftheSociety. SomeoftheJuniorFellowsgavemetheimpressionofbeingasomewhatpreciousgroupofyoungmen,asfarasmannerswereconcerned.Oxtoby,WillardQuine (reallyalogician),andIweretheonlymathematiciansamongthem.Amongthephysiciststherewereseveralwholaterbecameverywellknown,suchasJohn Bardeen,IvanGetting,andJimFisk.Amongthebiologists,IrememberRobertB.Woodward,thechemistwhofirstsynthesizedquinineandotherimportantbiological substances.PaulSamuelson,theeconomistwhoservedasadvisortoPresidentKennedy,wasthere;alsoIvarEinerson,agreatscholarinlinguistics;HenryGuerlach, whobecameahistorianofscience;andHarryLevin,inEnglishliterature.Levinwasratherproustianinhismanner.Helovedtoengageinsophisti­

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catedandwhatseemedtomeoccasionallyratherpreciousdiscussions.Anotherforeign­bornmemberwasGeorgeHanfmann,anarchaeologist.Hanfmannwas obviouslyaverylearnedperson,andIappreciatedhiserudition.WesharedthesamefondnessforGreekandLatinliterature. ThelogicianWillardQuinewasfriendlyandoutgoing.Hewasinterestedinforeigncountries,theircultureandhistory,andknewafewwordsofSlaviclanguages, whichheusedonmewithgreatgusto.Healreadyhadmadeareputationinmathematicallogic.Irememberhimasslim,dark­haired,dark­eyed—anintenseperson. Duringthepresidentialelectionof1936inwhichFranklinD.RooseveltdefeatedLandon,ImethimonthestairsofWidenerLibraryatnineinthemorning,after Roosevelt'slandslidevictory.WestoppedtochatandIaskedhim:"Well,whatdoyouthinkoftheresults?""Whatresults?"hereplied.''Thepresidentialelection,of course,"Isaid."WhoisPresidentnow?"heaskedcasually.Thiswascharacteristicofmanyinacademe.Ionceheardthat,duringCharlesW.Eliot'spresidencyat Harvard,avisitortohishousewastold,"ThePresidentisawayinWashingtontoseeMr.Roosevelt"!(ThiswasTheodoreRoosevelt.) IhadmymealsatAdamsHouse,andthelunchestherewereparticularlyagreeable.Wesatatalongtable—youngmenandsometimesgreatprofessors;the conversationswereverypleasant.Butoften,towardstheendofameal,oneaftertheotherwouldgulphiscoffeeandsuddenlyannounce:"Excuseme,I'vegottogo towork!"YoungasIwasIcouldnotunderstandwhypeoplewantedtoshowthemselvestobesuchhardworkers.Iwassurprisedatthislackofself­assurance,even onthepartofsomefamousscholars.LaterIlearnedaboutthePuritanbeliefinhardwork—oratleastinappearingtobedoinghardwork.Studentshadtoshowthat theywereconscientious;theolderprofessorsdidthesame.Thislackof'self­confidencewasstrangetome,althoughitwaslessobjectionablethantheEuropean arrogance.InPoland,peoplewouldalsopretendandfabricatestories,butin

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theoppositesense.Theymighthavebeenworkingfranticallyallnight,buttheypretendedtheyneverworkedatall.Thisrespectforworkappearedtomeaspartof thePuritanemphasisonactionversusthought,sodifferentfromthearistocratictraditionsofCambridge,England,forexample. TheSociety'sroomswereinEliotHouse.WeJuniorFellowswouldmeetthereonMondaysandFridaysforlunch,andforthefamousMonday­nightdinnerswhich gatheredJuniorandSeniorFellowstogetheraroundalongT­shapedtablewhichwassaidtobetheonefeaturedinOliverWendellHolmes'Autocratofthe Breakfast­Table.HendersonhadsecureditfromsomeHarvardstoreroom. PresidentLowellattendedalmosteveryMondaydinner.Hewasfondofre­creatingtheBattleofJutlandofWorldWarI,movingknivesandforksandsaltcellars aroundonthedinnertabletoshowthepositionsoftheBritishandGermanfleets.Fromtimetotimehewouldalsobetrayhisdoubtsandevenremorseaboutthe SaccoandVanzetticase.Hewouldrecountit—notsomuchtodefendbutrathertorestatethepositionofthecourtandthesubsequentlegalsteps.Hehadbeena memberofoneofthereviewcommittees. GoodFrenchBurgundiesorAlsatianwinesaccompaniedthemeals.TheseweretheprideandjoyofHenderson,whooncetoldmethatifheeverdeservedastatuein Cambridge,hewouldliketobeputinHarvardSquarewithabottleofwineinhishands,incommemorationofhishavingbeenthefirstpersontoobtainUniversity fundsforawinecellar.GeorgeHomans,oneoftheJuniorFellows,adescendantofPresidentJohnAdams,wasoneoftheyoungmenentrustedwiththeselectionand samplingofwines.IconsidereditagreatdistinctionwhenI,too,wasputonthewine­tastingcommitteeoftheSociety.Thiswasmyveryfirstadministrativejobin America!TheSocietyisstillverymuchalivetodayatHarvard,anditcontinuestoholditsMonday­nightdinnerswhereformerfellowsarealwayswelcome. In1936thedepressionappearedtobeending.Harvard

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Universityseemedrelativelyuntouchedbythiscataclysm.AfterthecolloquiumtalkIgavetherejustbeforemyappointmenttotheSocietyofFellows,Iremember ProfessorWilliamGrausteintellingmethatatHarvardtheprofessorshadnotfeltthedepressionatall.Thisleftmewonderingattheirlackofinvolvementinthegeneral problemsofthecountryorintheaffairsofMassachusettsorevenofCambridge.ItwasevidentthatcampuslifeinAmericameantatleastpartialisolationfromtherest ofsociety.ProfessorslivedalmostentirelyamongthemselvesandhadverylittlecontactwiththerestoftheprofessionalorcreativecommunityasinLwów.Thishad bothgoodandbadeffects:moretimeforscholarlywork,butverylittleinfluenceonthelifeofthecountryorviceversa.Aseveryoneknows,thingschanged somewhatafterWorldWarII.IntheKennedyadministration,forexample,Harvardianshadagreatdealtodowiththeaffairsofgovernmentandforatimethe influenceofscientistsbecameevenparamount. ActivitiesattheSocietyofFellowswereofcourseonlyonefacetofmylifeatHarvard.Ihadmanycontactswiththeyoungermembersofthefacultyattheuniversity andquiteoftensawandtalkedwiththeseniorprofessorsandwithG.D.Birkhoffhimself.HissonGarrett,atall,good­looking,andbrilliantmathematician,sometwo yearsyoungerthanI,becameafriend,andwesaweachothernearlyeveryday. EventhoughmembershipintheSocietydidnotrequireteachingofanykind,ProfessorGrausteinaskedmetoteachanelementaryundergraduatesectionofa freshmancoursecalledMath1A.(ItmayevenbethatthelatePresidentKennedywasforawhileastudentinthisclass.Irememberanamelikethatandsomeone sayingthattheyoungmanwasaratherremarkableperson.Helefttogoabroadinthemiddleoftheterm.YearslaterwhenImetPresidentKennedyIforgottoask himwhetherhehadreallytakenthatcourse.) Ihadgiventalksandseminars,butnotyettaughtaregularclass,andIfoundthisteachinginteresting.Theruleforyounginstructorswastofollowverycloselythe prescribed

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textbook.ApparentlyIdidnotdotoobadly,forinanevaluationofteachersthestudentnewspaperpraisedmeasaninterestinginstructor.Soonafterthebeginningof thecourse,G.D.Birkhoffcametoinspectmyperformance.PerhapshewantedtocheckmyEnglish.HesatatthebackoftheroomandwatchedasIexplainedto thestudentshowtowriteequationsofparallellinesinanalyticgeometry.ThenIsaidthatnextwewouldstudytheformulaeforperpendicularlines,which,Iadded, were"moredifficult."AfterthelectureBirkhoffcarnetomeandcommented,"You'vedoneverywell,butIwouldnothavesaidthatperpendicularlinesaremore difficult."IrepliedthatIbelievedonthecontrarythatstudentswouldrememberbetterthiswaythanifIsaideverythingwaseasy.Birkhoffsmiledatthisattemptat pedagogyonmypart.Ithinkhelikedmyindependenceandoutspokenways,andIsawhimratherfrequently. ShortlyafterIarrivedinCambridge,hehadinvitedmetodineathishouse.Itwasmyfirstintroductiontostrangedisheslikepumpkinpie.Afterdinner,whichwas pleasantenough,IgotreadytoleaveandG.D.tookmyovercoattohelpmeintoit.ThissortofcourtesywasunheardofinPoland;anoldermanwouldneverhave helpedamuchyoungerone.Irememberblushingcrimsonwithembarrassment. IfrequentlyatelunchwithhissonGarrett,andweoftentookwalkstogether.Wetalkedmuchaboutmathematicsandalsoindulgedintheusualgossipthat mathematicianslove.SurelyitisashallowthemetoevaluatehowgoodXorYis,butitisacharacteristicofourtribe.ThereadermayhavenoticedthatIpracticethis, too.Mathematicsbeingmoreinthenatureofanart,valuesdependonpersonaltastesandfeelingsratherthanonobjectivefactualnotions.Mathematicianstendtobe rathervain—thoughlesssothanoperatenorsorartists.Butaseverymathematicianknowssomespecialbitofmathbetterthananyoneelse,andmathissuchavast andnowmoreandmorespecializedsubject,someliketoproposelinearordersof"class"amongthebetter­

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knownonesandtocommentontheirrelativemerits.Onthewhole,itisaharmlessifsomewhatfutilepastime. IrememberthatattheageofeightornineItriedtoratethefruitsIlikedinorderof"goodness."Itriedtosaythatapearwasbetterthananapple,whichwasbetter thanaplum,whichwasbetterthananorange,untilIdiscoveredtomyconsternationthattherelationwasnottransitive—namely,plumscouldbebetterthannuts whichwerebetterthanapples,butappleswerebetterthanplums.Ihadfallenintoaviciouscircle,andthisperplexedmeatthatage.Mathematicians'ratingsare somethinglikethis. Manymathematiciansarealsosensitiveaboutwhattheyconsidertheirmostbeautifulmentaloffspring—resultsortheorems—andtheytendtobepossessiveabout them.Paradoxically,theyalsoshowatendencytoconsidertheirownworkasdifficultandotherworkaseasier.Thisisexactlyoppositeinotherfieldswherethe betteracquaintedoneiswithsomethingtheeasieritseems. Mathematiciansarealsopronetodisputes,andpersonalanimositiesbetweenthemarenotunknown.Manyyearslater,whenIbecamechairmanofthemathematics departmentattheUniversityofColorado,InoticedthatthedifficultiesofadministeringNpeoplewasnotreallyproportionaltoNbuttoN2.Thisbecamemyfirst "administrativetheorem."Withsixtyprofessorsthereareroughlyeighteenhundredpairsof'professors.Outofthatmanypairsitwasnotsurprisingthattherewere somewhosemembersdidnotlikeoneanother. AmongtheHarvardmathematiciansIknew,IshouldmentionHasslerWhitney,MarshallStone,andNorbertWiener.Whitneywasayoungassistantprofessor, interestingnotonlyasamathematician.Hewasfriendly,butrathertaciturn—psychologicallyofatypeoneencountersinthiscountrymorefrequentlythanincentral Europe—withwryhumor,shynessbutself­assurance,aprobitywhichshinesthrough,andacertaingeniusforpersistentanddeepfollow­throughinmathematics.

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MarshallStone,whomIhadmetwhenhecamethroughWarsawwithvonNeumannandBirkhoffin1935onthewaybackfromtheMoscowCongress,hadhada meteoriccareerattheuniversity,althoughhewasonlythirty­oneyearsold.Alreadyafullprofessor,hewasquiteinfluentialintheaffairsofthedepartmentandofthe universityforthatmatter.Hewroteaclassicwork,acomprehensiveandauthoritativebookonHilbertspace,aninfinitelydimensionalgeneralizationofthethree­ dimensionalorn­dimensionalEuclideanspace,mathematicallybasictomodernquantumtheoryinphysics.HewasthesonofHarlanStone,ChiefJusticeofthe SupremeCourt.ItissaidthathisfatherproudlysaidofMarshall'smathematicalachievements,"IampuzzledbuthappythatmysonhaswrittenabookofwhichI understandnothingatall." AndtherewasNorbertWiener!ImethimatacolloquiumtalkIgaveduringmyfirstyearatHarvard.Iwaslecturingonsomeproblemsoftopologicalgroups,and mentionedaresultIhadobtainedinPolandin1930ontheimpossibilityofcompletelyadditivemeasuredefinedinallsubsetsofagivenset.Wiener,whoalwayssatat lecturesinasemi­somnolentstateexceptwhenheheardhisname(atwhichhewouldsuddenlyjumpup,thensitbackinaverycomicalway)interruptedmetosay, "Oh!Vitalihasprovedsomethinglikethatalready."IrepliedthatIknewVitali'sresultandthatitwasmuchweakerthanminebecauseitrequiredanadditional property—namelyequalityforcongruentsets—whereasmyresultdidnotmakeanysuchpostulateandwasamuchstronger,purelysettheoreticalproof.Afterthe lecturehecametome,apologized,andagreedwithmystatements.Thiswasthebeginningofouracquaintance. IhadheardofWienerbeforethismeeting,ofcourse,notonlyabouthismathematicalwizardry,hisworkinnumbertheory,hisfamousTauberiantheorems,andhis workonFourierSeries,butalsoabouthiseccentricities.InPoland,IhadheardthroughJozefMarcinkiewiczabouthisbookwith

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PaleyonthesummabilityofFouriertransforms.RaymondPaley,oneofthemostpromisingandsuccessfulyoungEnglishmathematicians,diedinamountaineering accidentatatragicallyyoungage.MarcinkiewiczwasastudentofAntoniZygmund.HevisitedLwówasapost­doctoralfellowandpatronizedtheScottishCafé, wherewediscussedWiener'swork,sincehehadworkedintrigonometricseries,trigonometrictransforms,andsummabilityproblems.Marcinkiewicz,likePaley, whomheresembledingeniusandinmathematicalinterestsandaccomplishments,reportedlywaskilledwhileanofficerinthePolisharmyinthe1939campaignatthe beginningofWorldWarII. Absentmindedandotherworldlyinappearance,Wienerneverthelesscouldmakeanintuitiveappraisalofothers,andhemusthavebeeninterestedinme.Greatasthe differenceinagebetweenuswas(hisfortytomytwenty­sixyears),hewouldseekmeoutoccasionallyinmylittleapartmentinAdamsHouse,sometimeslateinthe evening,andproposeamathematicalconversation.Hewouldsay,"Let'sgotomyoffice,whereIcanwriteontheblackboard."Thissuitedmebetterthanstayingin myrooms,fromwhichitwouldhavebeendifficulttoputhimoutwithoutbeingrude.SohedrovemeinhiscarthroughdarkenedstreetstoMIT,openedthebuilding doors,turnedonthelight,andhestartedtalking.Afteranhourorso,althoughWienerwasalwaysinteresting,Iwouldalmostfallasleepandfinallymanagetosuggest thatitwastimetogohome. Wienerseemedchildishinmanyways.Beingveryambitiousabouthisplaceinthehistoryofmathematics,heneededconstantreassuranceabouthiscreativeability.I wasalmoststunnedafewweeksafterourfirstencounterwhenheaskedmepointblank:"Ulam!DoyouthinkIamthroughinmathematics?"Mathematicianstendto worryabouttheirdiminishingpowerofconcentrationmuchassomemendoabouttheirsexualpotency.Impudently,Ifeltastrongtemptationtosay"yes"asajoke, butrefrained;hewouldnothaveunderstood.Speakingofthatremark,"AmI

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through,"severalyearslateratthefirstWorldCongressofMathematiciansheldinCambridge,IwaswalkingonMassachusettsAvenueandsawWienerinfrontofa bookstore.Hisfacewasgluedtothewindowandwhenhesawme,hesaid,"Oh!Ulam!Look!Thereismybook!"Thenheadded,"Ulam,theworkwetwohave doneinprobabilitytheoryhasnotbeennoticedmuchbefore,butsee!Now,itisinthecenterofeverything."Ifoundthisdisarminglyandblessedlynaive. AnecdotesaboutWienerabound;everymathematicianwhoknewhimhashisowncollection.IwilladdmystoryofwhathappenedwhenIcametoMITasavisiting professorinthefallof1957.Iwasassignedanofficeacrossthehallfromhis.Ontheseconddayaftermyarrival,Imethiminthecorridorandhestoppedmetosay, "Ulam!Ican'ttellyouwhatIamworkingonnow,youareinapositiontoputasecretstamponit!"(ThispresumablybecauseofmypositioninLosAlamos.) Needlesstosay,Icoulddonosuchthing. Wieneralwayshadafeelingofinsecurity.BeforethewarheusedtotalkabouthispersonalproblemstoJ.D.Tamarkin,whowasagreatfriendofhis.Whenhewas writinghisautobiography,heshowedavoluminousmanuscripttoTamarkin.Tamarkin,whomIhadmetin1936andwithwhomIbecamequitefriendly,toldme aboutWiener'smanuscriptandhowinterestingitwas.ButhealsoexpressedtheopinionthatWienermightbesuedforlibelformanyofhisoutspokenstatements.He spokealmostwithdisbeliefaboutWiener'stextandhowhetriedtodissuadehimfrompublishingthebookinthatform.Whatfinallyappearedapparentlywas considerablytoneddownfromtheoriginalversion. AnothermemoryIhaveofWienerconcernshisaskingmetogowithhimtoSouthStationinBostontomeettheEnglishmathematicianG.H.Hardywhowascoming totheStatesforavisit.HeknewIhadmetHardyinEngland.Wecollectedanothermathematician,perhapsitwasNormanLevinson,andpickedupHardyatthe train.Wiener,who

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pridedhimselfonhisknowledgeoftheChinese,theircultureandeventheirlanguage,invitedeverybodyforlunchataChineserestaurant.Immediatelyhestarted talkingChinesetothewaiter,whoseemednottounderstandaword.Wienersimplyremarked,"HemustbefromthesouthanddoesnotspeakMandarin."(Wewere notquiteconvincedthatthiswasthecompleteexplanation.)Itwasaverypleasantlunchwithmuchmathematicaltalk.AndafterlunchWienerwhohadpickedupthe checkdiscoveredthathehadnomoney.Fortunatelywefoundthefewnecessarydollarsinourpockets.Wienerscrupulouslyreimburseduslater. ItwassaidthatWiener,althoughheconsideredhisprofessorshipatMITquitesatisfactory,wasverydisappointedthatHarvardneverofferedhimapost.Hisfather hadbeenaprofessoratHarvard,andNorbertwantedverymuchtofollowinhisfootsteps. AlthoughG.D.Birkhoffwasatleasttenyearshissenior,Wienerfeltarivalrywithhimandwantedtoequalorsurpasshiminmathematicalachievementandfame. WhenBirkhoff'scelebratedergodictheoremproofwaspublished,Wienertriedveryhardtogohimonebetterandproveanevenstrongertheorem.Hedidmanageit, butthestrengtheningwasnotassimpleorasfundamentalasG.D.'soriginalproof.Hereagainisanexampleofthecompetitivenatureofsomemathematiciansandthe sourcesoftheirambition. IthinkWienerhadmarveloustalentsasamathematician—thatisperspicacityandtechnicalgenius.Hehadasupremegeneralintelligencebut,inmyopinion,notthe sparkoforiginalitywhichdoestheunusualunrelatedtowhatothershavedone.Inmathematics,asinphysics,somuchdependsonchance,onapropitiousmoment. PerhapsvonNeumannalsolackedsomeofthe"irrational,"thoughwithhiswonderfulcreativity,hecertainlywenttoandachievedthelimitsofthe"reasonable." ThereareseveralwaysinwhichWienerandvonNeumannintersectedintheirinterestsandintheirfeelings

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aboutwhatwasimportantbothinpuremathematicsanditsapplications,butitisdifficulttocomparetheirpersonalities.NorbertWienerwasatrueeccentricandvon Neumannwas,ifanything,theopposite—areallysolidperson.Wienerhadasenseofwhatisworththinkingabout,andheunderstoodthepossibilitiesofusing mathematicsforseeminglymoreimportantandmorevisibleapplicationsintheoreticalphysics.HehadamarveloustechniqueforusingFouriertransforms,anditis amazinghowmuchthepowerofalgorithmsorsymbolismcouldaccomplish.Iamalwaysamazedhowmuchacertainfacilitywithaspecialandapparentlynarrow techniquecanaccomplish.Wienerwasamasteratthis.Ihaveseenothermathematicianswhocoulddothesameinamoremodestway.Forinstance,Steinhaus obtainedquitepenetratinginsightsintootherfields,andhisstudent,MarkKac,nowatRockefellerUniversity,surpassedhim.AntoniZygmundinChicago,another Pole,isamasterofthegreatfieldoftrigonometricseries.Severalofhisstudentshaveobtainedepoch­makingresultsinotherfields—forexample,PaulCohen,who didthisinsettheory,themostgeneralandabstractpartofmathematics. Idon'tthinkWienerwasparticularlyfondofcombinatorialthinkingorofworkingonfoundationsofmathematicologicalorsettheoreticalproblems.Atthebeginningof hiscareer,hemayhavegoneinthisdirection,butlaterheappliedhimselftootherfieldsandtonumbertheory. VonNeumannwasdifferent.Healsohadseveralquiteindependenttechniquesathisfingertips.(Itisraretohavemorethantwoorthree.)Theseincludedafacilityfor symbolicmanipulationoflinearoperators.Healsohadanundefinable"commonsense"feelingforlogicalstructureandforboththeskeletonandthecombinatorial superstructureinnewmathematicaltheories.Thisstoodhimingoodsteadmuchlater,whenhebecameinterestedinthenotionofapossibletheoryofautomata,and whenheundertookboththeconceptionandtheconstructionofelectroniccomputingmachines.Heattemptedtodefineandtopursuesomeofthe

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formalanalogiesbetweentheworkingsofthenervoussystemingeneralandofthehumanbrainitself,andtheoperationofthenewlydevelopedelectroniccomputers. Wiener,somewhathemmedinbythechildishnessandnaivetéofhispersonality,wasperhapspsychologicallyhandicappedbythefactthat,asachild,hisfatherhad pushedhimasaprodigy.VonNeumann,whoalsobeganratheryoung,hadamuchwiderknowledgeoftheworldandmorecommonsenseoutsidetherealmofpure intellect.Furthermore,WienerwasperhapsmoreinthetraditionoftalmudisticJudaicscholarship,eventhoughhisopinionsandbeliefswereverylibertarian.Thiswas quiteconspicuouslyabsentfromvonNeumann'smakeup. Johnny'soverwhelmingcuriosityincludedmanyfieldsoftheoreticalphysics,beginningwithhispioneeringwork—hisattempttoformarigorousmathematicalbasisfor quantumtheory.Hisbook,DieMathematischeGrundlagenderQuantumMechanik,publishedoverfortyyearsago,isnotonlyaclassic,butstillthe''bible"onthe subject.HewasespeciallyfascinatedbythepuzzlingroleoftheReynoldsnumberandtheseemingmysteryofsuddenonsetsofturbulenceinthemotionsoffluids.He haddiscussionswithWienerontheperplexingvaluesofthisnumberwhichis"dimensionless"—apurenumberexpressingtheratiooftheinertialforcestotheviscous forces.Itisoftheorderoftwothousand,alargenumber.Whyisthissoandnotaroundone,orten,orfifty?Atthattime,JohnnyandIcametotheconclusionthat actualdetailednumericalcomputationsofmanyspecialcasescouldhelpthrowlightonthereasonsforthetransitionfromalaminar(regular)toaturbulentflow. HetoldmeofanotherdiscussionhehadwithWienerandtheirdifferentpointsofview:Johnnyadvocated,inordertoestablishmodelsfortheworkingofthehuman brain,anumericaldigitalapproachthroughasequenceoftimesteps,whileWienerimaginedcontinuousor"hormonal"outlines.Thedichotomybetweenthesepointsof viewisstillofgreatinterestand,ofcourse,bynowhasbeentrans­

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formedanddeepenedbythegreaterknowledgeoftheanatomyofthebrainandbymoreworkinthetheoryofautomata. TherelationshipbetweenG.D.BirkhoffandvonNeumannwascurious.BirkhoffdidnotreallyhavecompleteadmirationfororappreciationofvonNeumann's genius.HeprobablycouldnotappreciatethemanykindsofmathematicsvonNeumannwaspursuing.Headmiredhistechnicalbrilliance,butG.D.'stastesweremore classical,inthetraditionofPoincaréandthegreatFrenchschoolofanalysis.VonNeumann'sinterestsweredifferent.Birkhoffhadambitionstoproducesomethingof greatimportanceinphysics,andhemadeafew,technicallyinterestingbutnotconceptuallyimportantcontributionstothegeneraltheoryofrelativity.Helectured severaltimesonsuchsubjectsinMexico,stimulatingasmallschoolofrelativiststhere.VonNeumann'sinterestslayinthefoundationsofthenewquantumtheory's morerecentdevelopments.Theirsweredifferencesofinterests,ofapproaches,andofvaluesystems.Birkhoffappreciatedprobingindepthmorethanexploringin breadth.VonNeumann,tosomeextent,didboth.Therewas,ofcourse,aboutaquarter­century'sdifferenceinagebetweenthem,aswellasinbackgroundandin upbringing.Also,vonNeumannneverquiteforgaveG.D.forhaving"scooped"himintheaffairof'theergodictheorem:VonNeumannhadbeenfirstinprovingwhat isnowcalledtheweakergodictheorem.Byasheervirtuosokindofcombinatorialthinking,Birkhoffmanagedtoproveastrongerone,and—havingmoreinfluence withtheeditorsoftheProceedingsoftheNationalAcademyofSciences—hepublishedhispaperfirst.ThiswassomethingJohnnycouldneverforget.Hesometimes complainedaboutthistome,butalwaysinamostindirectandobliqueway. InadditiontotheelementarymathematicscourseswhichItaughtduringmyfirstyearintheSociety,Iwasaskedtoaddadvancedcoursesgradually.Ilikedthis,for thebestwaytolearnasubjectistotrytoteachitsystematically.

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Thenonegetstherealpoints,theessentials.Onewasanimportantundergraduatecourseinclassicalmechanics,Math4ifIrememberitsformername.Anotherwas Math9,acourseonprobability. AtthetimeIhadnopreciseideawhatgradesmeant:A,B,C,D,orF.ButIhadrigidstandards.Irememberanotherwisequitegoodstudent,whoprotested receivingthegradeof"C."Someotherprofessorsintervened,butIstubbornly,perhapsfoolishly,stoodmyground.NowItendtobemorelenient,andwhenIgivea "C"or''D"thestudentsreallydeservean"F"orworse! Tamarkin,whowasaprofessoratBrownUniversity,askedmetoteachagraduatecourseinhisplacewhilehetookhissabbaticalleaveforaterm.Idecidedtogive thecourseonthetheoryoffunctionsofseveralrealvariables.Itincludedalotofnewmaterial—muchofitmyownrecentwork—andIwasratherproudofit.Every FridayIwenttoProvidencebytrain,taughtthecourse,spenttheweekendwithTamarkinathishome,returningtoCambridgeonSunday.WhenImentionedthe contentsofthecoursetoMazurwhenIwenthometoLwówforthelasttimeduringthesummerof1939,helikeditverymuch.Helikedthematerial,thewayitwas organized,andsaidhewouldlovetogivesuchacoursehimself,allofwhichpleasedmeandencouragedme. Tamarkinwasamostinterestingperson.Hewasofmediumheight,veryportly—Iwouldsaysomethirtypoundsoverweight.Hewasquitenearsighted,aconstant cigar­and­cigarette­smoker,andgenerallyextremelyjovial.AsIgottoknowhimbetter,Idiscoveredthewonderfulqualitiesofhismindandcharacter. BeforeWorldWarI,hehadwrittensomemathematicalresearchpapersontheworkofG.D.Birkhoffandevenimprovedsomeofthelatter'sresultsabit,whichled toacertainanimosityintheirrelations.YetwhenhecametotheUnitedStates,Birkhoffhelpedhimsecurehispositionat

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Brown,whichhadanotablemathematicsfaculty,includingJamesRichardson,RaymondC.Archibald,andothers.Richardsonwasagentlemanoftheoldschool. Archibaldwasaneminenthistorianofmathematics,whoestablishedthefamousmathematicallibraryofBrown,oneofthebestinthecountry. TamarkinwasinterestedinPolish­stylemathematicsandhadheardaboutsomeofmyresultsinthetheoryofBanachspaces.Hehadaqualitywhichperhapsonlya smallnumberofmathematicianspossess:hewasextremelyinterestedintheworksofothersandlessegocentricthanmost.Hewasalsointerestedinwhatwasgoing oninotherfieldsbesideshisown,whereasmostmathematician's—eventhebestones—areoftendeeplyimmersedintheirownworkanddonotpaymuchattention towhatothersaroundthemaredoing.Tamarkinbefriendedmeandencouragedmeinmywork. HewasRussian,notofJewishoriginexactly,butaKaraite.TheKaraiteswereasectofSemiticpeoplenotsubjecttotheusualrestrictionsonJewsinRussia,the reasonbeingthattheyclaimedtheywereabsentfromPalestinewhenJesuswascondemnedtodeath,andthisexemptedthem.Thisclaimwasacceptedbythe Russiangovernors.TheyalsohadsomethingincommonwiththeancientKhazars,peopleofamysterioussixth­orseventh­centurykingdominsouthernRussia,a pagantribewhosekingdecidedtoadoptanewreligion.HeselectedJudaismafterhavingaskedChristian,Moslem,andJewishrepresentativestoexplaintheirbeliefs. Tamarkinbelievedhewasoneoftheirdescendants.HehadescapedfromLeningradaftertheRussianRevolutioninamannernotunlikethatofGeorgeGamovsome tenyearslater—overtheiceofLakeLadogatoFinland. WhileIwasatHarvard,Johnnycametoseemeafewtimes,andIinvitedhimtodinnerattheSocietyofFellows.Wewouldalsotakeautomobiledrivesandtrips together

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duringwhichwediscussedeverythingfrommathematicstoliteratureandtalkedwithoutinterruptionwhilestillpayingattentiontooursurroundings.Johnnylikedthis kindoftravelverymuch. OnceatChristmastimein1937,wedrovefromPrincetontoDukeUniversitytoameetingoftheAmericanMathematicalSociety.Ontheway,amongotherthingswe discussedtheeffectthatthearrivalofincreasinglylargenumbersofrefugeeEuropeanscientistswouldhaveontheAmericanacademicscene.Westoppedataninn wherewefoundafolderdescribingalocalIndianChief,Tomo­Chee­Chee,whoapparentlyhadbeenunhappyaboutthearrivalofwhitemen.Asanillustrationofour frequentlylinguisticandphilologicaljokes,IaskedhimwhyitwasthatthePilgrimshad"landed"whilethepresentEuropeanimmigrantsandscientificrefugeesmerely "arrived."Johnnyenjoyedtheimpliedcontrastandusedthisinothercontextsasanexampleofanimpliedvaluejudgment.WealsolikenedG.D.Birkhoff'sincreasing qualmsabouttheforeigninfluencetotheIndianChief's.Continuingourdrive,wemanagedtoloseourwayacoupleoftimesandjokedthatitwasChiefTomo­Chee­ Cheewhohadmagicallyassumedtheshapeoffalseroadsignstoleadusastray. ThiswasthefirsttimeIvisitedtheSouth,andIwasmuchtakenbythedifferenceinatmospherebetweenNewYork,NewEngland,andthesouthernstates.I rememberafeelingof"déjàvu":themorepolishedmanners,themoreleisurelypaceoflife,andtheelegantestates.Somethingseemedfamiliar,andIwonderedwhatit was.SuddenlyIaskedmyself'ifitcouldbetheremnantsofthepracticeofslavery,whichremindedmeofthetracesoffeudalismstillvisibleinthecountrylifeof Poland.Iwasalsosurprisedtoseesomanyblackpeople,andtheirlanguageintriguedme.Atagasstation,oneoftheNegroattendantssaid,"Whatwouldyoulike now,Captain?"IaskedJohnny,"DoeshethinkImightbeanofficerandcallsmeCaptainasacompli­

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ment?"Similarly,thefirsttimeIheardmyselfcalled"Doc,"IwonderedhowtheporterknewthatIhadaDoctor'sdegree! AswepassedthebattlefieldsoftheCivilWar,Johnnyrecountedthesmallestdetailsofthebattles.Hisknowledgeofhistorywasreallyencyclopedic,butwhathe likedandknewbestwasancienthistory.HewasagreatadmireroftheconciseandwonderfulwaytheGreekhistorianswrote.HisknowledgeofGreekenabledhim toreadThucydides,Herodotus,andothersintheoriginal;hisknowledgeofLatinwasevenbetter. ThestoryoftheAthenianexpeditiontotheislandofMelos,theatrocitiesandkillingsthatfollowed,andthelengthydebatesbetweentheopposingpartiesfascinated himforreasonswhichIneverquiteunderstood.HeseemedtotakeaperversepleasureinthebrutalityofacivilizedpeopleliketheancientGreeks.Forhim,Ithinkit threwacertainnot­too­complimentarylightonhumannatureingeneral.Perhapshethoughtitillustratedthefactthatonceembarkedonacertaincourse,itisfatedthat ambitionandpridewillpreventapeoplefromswervingfromachosencourseandinexorablyitmayleadtoawfulends,asintheGreektragedies.Needlesstosaythis propheticallyanticipatedthevasterandmoreterriblemadnessoftheNazis.Johnnywasverymuchawareoftheworseningpoliticalsituation.InaPythianmanner,he foresawthecomingcatastrophe. ItwasduringthistripalsothatforthefirsttimeIsensedthathewashavingproblemsathome.Heexhibitedacertainrestlessnessandnervousnessandwould frequentlystoptotelephonetoPrinceton.Oncehecamebacktothecarverypaleandobviouslyunhappy.Ilearnedlaterthathehadjustfoundoutthathismarriage toMariettawasdefinitelybreakingup.Shewouldleavehimshortlythereaftertomarryayoungerphysicist,oneofthefrequentguestsatthenumerouspartieswhich thevonNeumannsgaveinPrinceton.

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OnthewaybackfromthemeetingIposedamathematicalproblemabouttherelationbetweenthetopologyandthepurelyalgebraicpropertiesofastructurelikean abstractgroup:whenisitpossibletointroduceinanabstractgroupatopologysuchthatthegroupwillbecomeacontinuoustopologicalgroupandbeseparable? "Separable"meansthatthereexistsacountablenumberofelementsdenseinthewholegroup.(Namely,everyelementofthegroupcanbeapproximatedbyelements ofthiscountableset.)Thegroup,ofcourse,hastobeofpowercontinuumatmost—obviouslyanecessarycondition.Itwasoneofthefirstquestionswhichconcern therelationbetweenpurelyalgebraicandpurelygeometricortopologicalnotions,toseehowtheycaninfluenceordetermineeachother. Weboththoughtaboutwaystodoit.Suddenly,whilewewereinamotelIfoundacombinatorialtrickshowingthatitcouldnotbedone.Itwas,ifIsaysomyself, ratheringenious.IexplainedittoJohnny.AswedroveJohnnylatersimplifiedthisproofinthesensethathefoundanexampleofacontinuumgroupwhichiseven Abelian(commutative)andyetunabletoassumeaseparabletopology.Inotherwords,thereexistabstractgroupsofpowercontinuuminwhichthereisnopossible continuousseparabletopology.Whatismore,thereexistsuchgroupsthatareAbelian.Johnny,wholikedverbalgamesandtoplayonwords,askedmewhattocall suchagroup.Isaid,"nonseparabilizable."Itisadifficultwordtopronounce;onandoffduringthecarrideweplayedatrepeatingit. Mathematicianshavetheirownbrandof"in"humorlikethis.Generallyspeaking,theyareamusedbystoriesinvolvingtrivialityofidentityoftwodefinitionsor "tautologies."Theyalsolikejokesinvolvingvacuoussets.Ifyousaysomethingwhichistrue"invacuo,"thatistosay,theconditionsofthestatementarenever satisfied,itwillstrikethemashumorous.Theyappreciateacertaintypeoflogicalnonsequiturorlogicalpuzzle.Forinstance,thestoryoftheJewishmotherwhogives apresentoftwotiestoherson­in­law.

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Thenexttimesheseeshim,heiswearingoneofthem,andsheasks,"Youdon'tliketheotherone?" SomeofvonNeumann'sremarkscouldbedevastating,eventhoughthesarcasmwasofanabstractnature.EdCondontoldmeinBoulderofatimehewassitting nexttoJohnnyataphysicslectureinPrinceton.Thelecturerproducedaslidewithmanyexperimentalpointsand,althoughtheywerebadlyscattered,heshowedhow theylayonacurve.AccordingtoCondon,vonNeumannmurmured,"Atleasttheylieonaplane." Somepeopleexhibitanabilitytorecallstoriesandtellthemtoothersonappropriateoccasions.Othershavetheabilitytoinventthembyrecognizinganalogiesof situationsorideas.Athirdgrouphastheabilitytolaughandenjoyotherpeople'sjokes.Isometimeswonderiftypesofhumorcouldbeclassifiedaccordingto personality.Myfriendsandcollaborators,C.J.EverettintheUnitedStatesandStanislawMazurinPoland,eachhadawrysenseofhumor,andphysicallyandin theirhandwritingtheyalsoresembleeachother. GenerallyvonNeumannpreferredtotellstorieshehadheard;Iliketoinventthem."Ihavesomewit;itisatremendousquality,"mywifesaysIoncetoldher.When shepointedoutthatIwasbragging,Ipromptlyadded,"True.Myfaultsareinfinite,butmodestypreventsmefrommentioningthemall." Inadditionto"in"jokes,mathematiciansalsopracticeaformof"in"language.Forexample,theyusetheword"trivial."Itisanexpressiontheyareveryfondof,but whatdoesitreallymean?Easy?Simple?Banal?AcolleagueofmyfriendGian­CarloRotaoncetoldhimthathedidnotliketeachingcalculusbecauseitwasso trivial.Yet,isit?Simpleasitis,calculusisoneofthegreatcreationsofthehumanmind,withbeginningsdatingbacktoArchimedes.Itwas"invented''byNewtonand Leibnitz,andamplifiedbyEuler,Lagrange,andothers.Ithasabeautyandanimpor­

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tancegoingfarbeyondmostofthemathematicsofourpresentculture.Sowhatis"trivial"?CertainlynotCantor'sgreatsettheory,technicallyverysimple,butdeep andwonderfulconceptuallywithoutbeingdifficultorcomplicated. Ihaveheardmathematicianssneeratthespecialtheoryofrelativity,callingitnothingbutatechnicallytrivialquadraticequationandafewconsequences.Yetitisone ofthemonumentsofhumanthought.Sowhatis"trivial"?Simplearithmetic?Itmaybetrivialtous,butisittothethird­gradechild? Letusconsidersomeotherwordsmathematiciansuse:whatabouttheadjective"continuous"?Outofthisonewordcamealloftopology.Topologymaybe consideredasabigessayontheword"continuous"inallitsramifications,generalizations,andapplications.Trytodefinelogicallyorcombinatoriallyanadverblike "even"or"nevertheless."Ortakeanordinarywordlike''key,"asimpleobject.Yetitisanobjectfarfromeasytodefinequasi­mathematically."Billowing"isamotion ofsmoke,forexample,inwhichpuffsareemittedfrompuffs.Itisalmostascommoninnatureaswavemotion.Suchawordmaygiverisetoawholetheoryof transformationsandhydrodynamics.Ioncetriedtowriteanessayonthemathematicsofthree­dimensionalspacethatwouldimitateit. WereIthirtyyearsyoungerImighttrytowriteamathematicaldictionaryabouttheoriginsofmathematicalexpressionsandconceptsfromcommonlyusedwords, imitatingthemannerofVoltaire'sDictionnairePhilosophique.

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Chapter6 TransitionandCrisis 1936–1940 Eachsummerbetween1936and1939,IreturnedtoPolandforafullthreemonths.Thefirsttime,afteronlyafewmonths'stayinAmerica,Iwassurprisedthatstreet carsran,electricityandtelephonesworked.IhadbecomeimbuedwiththeideaofAmerica'sabsolutetechnologicalsuperiorityandunique"know­how."Mymain emotionalreactionswere,ofcourse,relatedtoreunionwithmyfamilyandfriends,andthefamiliarscenesofLwów,followedbyalongingtoreturntothefreeand hopeful"open­ended"conditionsoflifeinAmerica.Tosimplifyadescriptionofthesecomplicatedfeelings:inMayIstartedcountingthedaysandweeksleftbefore returningtoEurope,thenafterafewweeksinPolandIwouldcountthedaysimpatientlybeforeIwouldreturntoAmerica. MostmathematiciansremainedinLwówduringthe

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summer,andoursessionsinthecoffeehousesandmyownpersonalcontactswiththemcontinueduntiltheoutbreakofWorldWarII.Asbefore,Iworkedwith BanachandMazur.Twice,whileBanachwasspendingafewdaysinSkoleorinnearbyvillagesintheCarpathianmountains,someseventymilessouthofLwów,I visitedhim.Iknewtheseplacesfrommychildhood.Banachwasworkingonsomeofhistextbooks,buttherewasalwayslotsoftimetositinacountryinnand discussmathematicsand"therestoftheuniverse,"anexpressionwhichwasdeartovonNeumann.ThelasttimeIsawBanachwasinlateJulyof1939attheScottish Café.WediscussedthelikelihoodofwarwithGermanyandinscribedafewmoreproblemsintheScottishBook. Inthesummerof1937,BanachandSteinhausaskedmetoinvitevonNeumanntocomeandgivealectureinLwów.HearrivedfromBudapestandspentseveral daysamongus.Hegaveanicelecture,andIbroughthimtotheCaféseveraltimes.HejottedsomeproblemsintheScottishBook,andwehadsomeverypleasant discussionswithBanachandseveralothers. ItoldBanachaboutanexpressionJohnnyhadonceusedinconversationwithmeinPrincetonbeforestatingsomenon­Jewishmathematician'sresult,"DieGoim habendenfolgendenSatzbewiesen"(Thegoyshaveprovedthefollowingtheorem).Banach,whowaspuregoy,thoughtitwasoneofthefunniestsayingshehadever heard.Hewasenchantedbyitsimplicationthatifthegoyscoulddoit,JohnnyandIoughttobeabletodoitbetter.Johnnydidnotinventthisjoke,buthelikeditand westartedusingit. IshowedJohnnythecity.Ihadexperienceinshowingittoforeignmathematicians;whenIwasonlyafreshman,becauseIcouldspeakEnglish,Kuratowskihad assignedtomethejobofshowingthetowntotheAmericantopologistAyres.IalsohadtoescortEdwardCzech,G.T.Whyburn,andseveralothersaroundLwów duringtheirPolishvisits.

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JohnnywasmuchinterestedinLwówandsurprisedatthenineteenth­centuryappearanceofthecenteroftownanditsmanyrelicsofthefifteenth,sixteenthand seventeenthcenturies.BothHungaryandPolandwerestillsemi­feudalinsomerespects.Thereweremanypicturesquepartsoftown,whereoldhousesleaned towardseachother,andcrookednarrowcobbledstreets.InonelittlestreetintheGhetto,black­marketoperationsincurrencywereconductedopenly.Butcher shopsinthesuburbshadsidesofbeefhangingexposedtofullview.Therewerestillhorse­drawncarriagesandelectricstreet­carlines.Taxiswerenotnumerous,and eveninthelatenineteen­thirtiesonecouldtakeahorse­drawn"fiacre,"usuallypulledbytwohorses.WhenIfirstarrivedinNewYork,Iwassurprisedtoseethe shabbyoldfiacresinfrontofthebestFifthAvenuehotelswithonlyonepoorhorsetopullthem. WevisitedanArmenianchurchwithfrescoesbyJanHenrykRosen,acontemporaryPolishartistnowintheUnitedStates.WealsowentintoalittleRussian Orthodoxchurch,wherewewerebothshockedbythesightofacorpseinahalf­opencoffinabouttobeburiedaccordingtoRussianritual.ItwasthefirsttimeIhad everseenadeadperson. Johnnyalsocametoourhouse.Hemetmyparents—mymother,whowastodiethefollowingyear,andmyfather,whohadheardsomuchabouthimfromme.I tookhimtoseemyfather'soffices,whichwereinadifferentpartofourbighouseonKosciuskoStreet. Johnnyalreadyknewsomeofmyfamily.Anauntofmine,thewidowofmyfather'sbrotherMichael,hadmarriedaHungarianfinancierbythenameofArpadPlesch. VonNeumannknewthePlesches.Arpad'sbrotherJanoswasEinstein'sphysicianinBerlin.Arpadwasanimmenselyrichfinancier,butarathercontroversialfigure. Myauntwaswealthytoo,aremarkablewoman,descendedfromafamousfifteenth­centuryPraguescholarnamedCaro.InIsraelmanyyearslater,whileIwas visitingthetownofSafedwithvon

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Kárman,anoldOrthodoxJewishguidewithearlocksshowedmethetombofCaroinanoldgraveyard.WhenItoldhimthatIwasrelatedtoaCaro,hefellonhis knees—thatcostmeatripletip.ThePleschestraveledfrequentlyandlivedofteninParis.Ivisitedthemthereonmytripin1934.Myaunt'sfirsthusband,Michael Ulam,myuncle,wasburiedinMonteCarlo,andmyaunt,whoisnowdeadtoo,isalsoburiedthereinafantasticmarblemausoleumintheCatholiccemetery.Aunt CarowasdirectlyrelatedtothefamousRabbiLoewofsixteenth­centuryPrague,who,thelegendsays,madetheGolem—theearthengiantwhowasprotectorofthe Jews.(Once,whenImentionedthisconnectionwiththeGolemtoNorbertWiener,hesaid,alludingtomyinvolvementwithLosAlamosandwiththeH­bomb,"Itis stillinthefamily!")Somuchforthefamily'srichconnections. AstorywhichJohnnyandIlikedtotelleachother,thoughIdonotrememberwhoproposeditfirst,wasaquotefromoneofthoserichuncles,whousedtosay: "Reichseinistnichtgenug,manmusstauchGeldinderSchweizhaben!"(Itisnotenoughtoberich,onemustalsohavemoneyinSwitzerland.) Inthesummerof1938,itwasvonNeumann'sturntoinvitemetoBudapest.ItraveledbytrainviaCracow,andwentdirectlytohishouse.(Ithinktheaddresswas 16AranyJánosStreet.)HehadreservedaroomformeattheHotelHungaria,thebesthotelintownatthattime.Itwasattheendofanarrowlittlestreet,sonarrow infact,thattherewasarevolvingplatformattheendforcarstoturnaroundon,asforlocomotivesinaroundhouse. JohnnyshowedmeBudapest.ItwasabeautifulcitywiththehousesofParliamentandthebridgesontheriver.Afterdinnerathishouse,whereImethisparents,we wenttonightclubsanddiscussedmathematics!Johnnywasalonethatyear,hismarriagewasbreakingupandMariettahadremainedinAmerica. Thenextdaysittingina"Konditorei,"talking,joking,

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andeating,wesawanelegantlydressedladygoingby.Johnnyrecognizedher.Sheentered,andtheyexchangedafewwords.Aftersheleft,heexplainedthatshewas anoldfriend,recentlydivorced.Iaskedhim,"Whydon'tyoumarrythedivorcée?"Perhapsthisimplantedthethoughtinhismind.Thenextyeartheyindeedwere married.HernamewasKlaraDan.Webecameverygoodfriendslateron.JohnnyandKlari,asshewasknowntoherfriends,weremarriedinBudapestandshe movedtoPrincetoninthelatesummerorfallof1939.Klariwasamoodyperson,extremelyintelligent,verynervous,andIoftenhadthefeelingthatshefeltthat peoplepaidattentiontohermostlybecauseshewasthewifeofthefamousvonNeumann.Thiswasnotreallythecase,forshewasaveryinterestingpersoninher ownright.Nevertheless,shehadtheseapprehensions,whichmadeherevenmorenervous.Shehadbeenmarriedtwicebefore(andmarriedafourthtimeaftervon Neumann'sdeath).Shediedin1963intragicandmysteriouscircumstances.AfterleavingapartygiveninhonorofNobelPrize­winnerMariaMayer,shewasfound drownedonthebeachatLaJolla,California. JohnnytookmealsotoapleasantmountainresortcalledLillafüredtovisithisformerprofessors,LeopoldFejerandFrederickRiesz,whowerebothpioneer researchersinthetheoryofFourierseries.Lillafüred,aboutahundredmilesfromBudapest,wasaresortwithluxuriouscastle­likebighotels.FejerandRieszwerein thehabitofsummeringthere.FejerhadbeenJohnny'steacher.Rieszwasoneofthemostelegantmathematicalwritersintheworld,knownforhisprecise,concise, andclearexpositions.Hewasoneoftheoriginatorsofthetheoryoffunctionspaces—ananalysiswhichisgeometricalinnature.Hisbook,FunctionsofReal Variables,isaclassic.Weweretowalkintheforest,butinthemorningbeforethewalkJohnnysaidwehadtowaituntilthemasterhadhisinspiration.Bythishe meantthedailyphysiologicalnecessities—notspiritualones,whichhadtobemetwithapre­breakfastsipofbrandybeforethe

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daycouldbegin!Wehadanicediscussion.Ofcoursethetalkalsoconcernedtheworldsituationandthelikelihoodofwar. IreturnedtoPolandbytrainfromLillafüred,travelingthroughtheCarpathianfoothills.Ihadtochangetrainsseveraltimesandremembersittingforawhileonanopen flatcarwithmylegsdanglingoverthesideaswewentthroughsmallvillagesnamedSatorolia­UjhelyandMunkaczewo.Thiswholeregiononbothsidesofthe CarpathianMountains,whichwaspartofHungary,Czechoslovakia,andPoland,wasthehomeofmanyJews.JohnnyusedtosaythatallthefamousJewishscientists, artists,andwriterswhoemigratedfromHungaryaroundthetimeofthefirstWorldWarcame,eitherdirectlyorindirectly,fromtheselittleCarpathiancommunities, movinguptoBudapestastheirmaterialconditionsimproved.ThephysicistI.I.RabiwasborninthatregionandbroughttoAmericaasaninfant.Itwillbeleftto historiansofsciencetodiscoverandexplaintheconditionswhichcatalyzedtheemergenceofsomanybrilliantindividualsfromthatarea.Theirnamesaboundinthe annalsofmathematicsandphysicsoftoday.Johnnyusedtosaythatitwasacoincidenceofsomeculturalfactorswhichhecouldnotmakeprecise:anexternal pressureonthewholesocietyofthispartofCentralEurope,afeelingofextremeinsecurityintheindividuals,andthenecessitytoproducetheunusualorelseface extinction.TomethepicturewasthatoftheRomanpoetVirgil,describingtheflood:"Inthebigwhirlpoolthereappearonlyafewremainingswimmers,"surviving throughintellectualfrenzyandstrenuousandvigorouswork.AjocularversionofsurvivalisastoryItoldtoJohnnywhomanufacturedmanyvariations.AlittleJewish farmboynamedMoysheWasserpissemigratedtoViennaandbecameasuccessfulbusinessman.HechangedhisnametoHerrWasserman.GoingontoBerlinand toevengreatersuccessandfortune,hebecameHerrWasserstrahl,thenvonWasserstrahl.NowinParisandstillmoreprosperous,heisBaronMauricedela Fontaine.

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EugeneWignerisoneofthefamousscientistsfromBudapest.HeandJohnnywereschoolfriendsandstudiedtogetherforatimeinZürich.Johnnytoldmeanice storyfromthosedays:EugeneandJohnnywantedtolearntoplaybilliards.Theywenttoacaféwherebilliardswereplayedandaskedanexpertwaiterthereifhe wouldgivethemlessons.Thewaitersaid,"Areyouinterestedinyourstudies?Areyouinterestedingirls?Ifyoureallywanttolearnbilliards,youwillhavetogiveup both."JohnnyandWignerheldashortconsultationanddecidedthattheycouldgiveuponeortheotherbutnotboth.Theydidnotlearntoplaybilliards. VonNeumannwasprimarilyamathematician.Wignerisprimarilyaphysicist,butalsohalfamathematicianandbrilliantuserofmathematics,avirtuosoof mathematicaltechniquesinphysics.Iwouldaddherethatherecentlypublishedaninterestingarticleontheaprioriunexpectedeffectivenessofmathematicsinphysics. VonNeumann'sbookonthefoundationsofquantumtheoryhadmorephilosophicalandpsychologicalmeaningratherthandirectapplicationsintheoreticalphysics. Wignermademanyconcretecontributionstophysics,perhapsnothingquiteasoverwhelmingasEinstein'sideasofrelativity,butmanyimportantspecifictechnical achievementsandalsosomethingrathergeneral—namely,theveryfundamentalroleofgrouptheoreticalprinciplesinthephysicsofquantumtheoryandthephysicsof elementaryparticles. WhenvonNeumanndied,WignerwroteabeautifulobituaryarticleinwhichhedescribedthedeepdespairthatcameuponJohnnywhenheknewthathewasdying, foritwasimpossibleforhimtoimaginethathewouldstopthinking.ForWigner,vonNeumannandthinkingweresynonymous. AfterthisvisittoBudapestitwastimetopreparemyreturntoHarvard.IhadtogototheAmericanconsulateinWarsaweachsummerIwasinPolandtoapplyfora newvisitor'svisainordertoreturntotheUnitedStates.Finally,the

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consulsaidtome,"Insteadofcominghereeverysummerforanewvisa,whydon'tyougetanimmigrationvisa?"ItwasluckythatIdid,forjustafewmonthslater thesebecamealmostimpossibletoobtain. TwiceImadeplanstotravelacrosstheAtlanticwithJohnny.TheyearhismarriagewithMariettabrokeup,wewenttoEuropetogether.IjoinedhimontheGeorgic, asmallCunardboatwhichtookaweektocross.Johnnyalwaystraveledfirstclass,soIwentfirstclassalso,althoughusuallyItraveledtourist.Asalways,we discussedalotofmathematics.WeflirtedwithayoungwomannamedFlatau,whomwefoundratherattractive.Thedayafterwemether,Iaskedhim:"Haveyou solvedtheproblemofFlatau?"Helikedtheplayonwords.InmathematicsthereisthefamousproblemofPlateau:givenacurveorawireinspace,theproblemisto findasurfacesuchthatthewireisitsboundaryandofminimalsurfacearea.Itcanbedemonstratedwithsoapbubbles.Ifyouimmerseaclosedcurvedwireinasoap solution,yougetsomenicesurfacesspanningthem.ThemanwhofirstformulatedandstudiedthismathematicallywasPlateau. In1939,mythree­yearappointmentattheSocietyofFellowswasexpiring.Unfortunately,itwasnotrenewablebecauseIhadpassedtheupperagelimit.Thanksto G.D.Birkhoff,IreceivedanextensionofthestayatHarvardintheformofalectureshipinthemathematicsdepartment.Morepermanentprospectswerenot promising;itseemedtherewerenovacantassistantprofessorships.BecauseoftheinfluxoflargenumbersofGermanandCentralEuropeanscientists,andinspiteof Johnny'seffortsonmybehalf,thingsdidnotlookmuchbetterinPrinceton.Thus,IreturnedtoAmericawithanassuredpositionforonlyonemoreyear,accompanied thistimebymyyoungerbrotherAdam,whowasnotquiteseventeen.WithAdaminmycharge,asenseofresponsibilitycameoverme.Ourmotherhaddiedtheyear before,andthefeelingofimpendingcrisishadconvincedmyfatherthatAdamwouldbesaferin

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Americawithme.Whenhetriedtoapplyforavisa,theAmericanconsulinWarsawseemedtohavereservations.ItisonlywhenIprovedthatIwaslivingand teachingintheUnitedStatesthatheagreedtoissuehimastudentvisa. OurfatheranduncleSzymonaccompaniedustoGdynia,aPolishportontheBalticSea,toseeusoffonthePolishlinerBatory.Thiswasthelasttimewewereto seeeitherofthem. WewereatseawhentheannouncementofthepactbetweenRussiaandGermanycameovertheship'sradio.Inastateofstrangeagitation,ItoldAdamuponhearing thenews,"ThisistheendofPoland."Onamapintheship'ssalon,IdrewalinethroughthemiddleofPoland,saying,Cassandra­like,"Itwillbedividedlikethat."We were,tosaytheleast,shakenandworried. Atdinneronthefirstnight,IsuddenlynoticedAlfredTarskiinthediningroom.Ihadnoideahewasontheboat.Tarski,famouslogicianandlecturerinWarsaw,told ushewasonhiswaytoaCongressontheUnityofPhilosophyandSciencetobeheldinCambridge.ItwashisfirsttriptoAmerica.Weateatthesametableand spentagooddealoftimetogether.Istillhaveanoldshipboardphotograph,whichshowsAdam,Tarski,andmedressedindinnerjacketsreadyforthegayAmerican sociallife.Heintendedtostayonlyacoupleofweeksandwastravelingwithasmallsuitcaseofsummerclothes.Becausethewarbrokeoutshortlyafterwelanded, hefoundhimselfstrandedintheUnitedStateswithoutmoney,withoutajob,andwithhisfamily—awifeandtwosmallchildren—inWarsaw.Forsometimehewasin themostprecariousandterriblesituation. Adamwasfrightenedandnervouswhenwelanded,ayoungboyabroadforthefirsttimeandawayfromfamiliarsurroundings.Johnnyhadcometomeetusatthe boat;onseeingAdam,heasked,outofhishearing,"Whoisthisfellow?"Hehadnotheardmyintroductionandwassurprised.Thereisadifferenceofthirteenyears betweenmy

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brotherandme,andwedonotlookatallalike.AdamistallerthanI,straight,blond,andwithapinkcomplexion.Iamsomewhatdarkandstockier.Inappearance heresemblessomeofouruncles,whileIlookmorelikeourmother.Atthepier,Johnnyappearedveryagitated.PeopleintheUnitedStateshadamuchclearerand morerealisticviewofeventsthanwehadhadinPoland.Forexample,whenitwastimetoobtainanexitvisafromPoland,becauseIwasinthePolisharmyreserveI hadtofirstsecurethepermissionofthearmytoleavethecountry.TheofficerinchargeaskedcasuallywhyIwantedtogoabroadandraisednofurtherquestion whenItoldhimofmylecturingengagementinAmerica.AsarulepeopleinPolandhadnotfelttheimminenceofwar,butratheracontinuationofthestateofcrisis, similartotheoneinMunichtheyearbefore. WestayedinNewYorkafewdays,visitingcousins—thepainterZygmundMenkèsandhiswife—andafriendofthefamilywhohadgivenadditionalfinancial guaranteesforAdam,whohadastudentvisa.Actually,theplanwasthatmybrotherwouldreceivemonthlychecksfromhomeviaouruncle'sbankinEngland.We alsosawyoungMr.Loeb,anacquaintanceofmycousinAndrzej,andwhenItalkedtohimonthetelephone,heasked:''WillPolandgivein?"IrepliedthatIwas prettysureitwouldneversurrender,andtherewasgoingtobeawar. IleftAdaminNewYorktogowithJohnnytoVeblen'ssummerplaceinMaine.Thoughweweregoneonlytwoorthreedays,Adamwasveryunhappywithmefor havinglefthim.OnthewaytoVeblen's,wediscussedsomemathematicsasusualbutmostlytalkedaboutwhatwasgoingtohappeninEurope.Wewereboth nervousandworried;weexaminedallpossiblecourseswhichawarcouldtake,howitcouldstart,when.AndwedrovebacktoNewYork.Thesewerethelastdays ofAugust. AdamandIwerestayinginahotelonColumbusCircle.Itwasaveryhot,humid,NewYorknight.Icouldnotsleepverywell.Itmusthavebeenaroundoneortwo inthemorn­

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ingwhenthetelephonerang.Dazedandperspiring,veryuncomfortable,Ipickedupthereceiverandthesomber,throatyvoiceofmyfriendthetopologistWitold Hurewiczbegantorecitethehorribletaleofthestartofwar:"Warsawhasbeenbombed,thewarhasbegun,"hesaid.ThisishowIlearnedaboutthebeginningof WorldWarII.Hekeptdescribingwhathehadheardontheradio.Iturnedonmyown.Adamwasasleep;Ididnotwakehim.Therewouldbetimetotellhimthe newsinthemorning.OurfatherandsisterwereinPoland,soweremanyotherrelatives.Atthatmoment,Isuddenlyfeltasifacurtainhadfallenonmypastlife, cuttingitofffrommyfuture.Therehasbeenadifferentcolorandmeaningtoeverythingeversince. OnthewaytoCambridge,IaccompaniedAdamtoBrownUniversityinProvidence,registeredhimasafreshman,andintroducedhimtoafewofmyfriends, includingTamarkinandhisson.HisEnglishwasquitegood,andhedidnotseemtomindbeingleftaloneincollege. Ibecameacompulsivebuyerofnewspapers,alltheextraeditionseveryhour,andeasilywentthrougheightortenpapersadaylookingfornewsofLwów,ofthe militarysituation,andoftheprogressofbattles.EarlyinSeptemberIsawintheBostonGlobealargephotographofAdamsurroundedbyotheryoungfreshmenat Brown.Itwascaptioned,"Wonderswhetherhishomewasbombed." Fromthestart,Adamdidverywellinschool;afewmonthslater,hewasabletoobtainatuitionwaiver.Neverthelesswewerefindingourselvesinseverefinancial straits.TheproposedincomefromBritainhadbeenfrozen—theEnglishgovernmentstoppedalloutgoingmoney,andmysalaryasalectureratHarvardwashardly sufficienttoputayoungerbrother(whowasnotallowedtoworkbecausehewasonastudentvisa)throughcollege.Onprevioustrips,Ihadneverthoughtof transferringfundsorpropertyfromPoland.Nowitcouldnolongerbedone.IwenttoseeadeanoftheCollegeandexplainedmysituation.HisnamewasFerguson, andIfearedthatwithsuchaScottishname

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hewouldberatherparsimonious.Fortunatelyhewasnot.ItoldhimthatifIcouldnotgetalittlemoreassistancefromtheUniversity,Iwouldbeforcedtoleavemy academiccareerandlookforsomeothermeansofsupport.Hewassympatheticandwasabletofindtwoorthreehundreddollarsmorefortheyear,whichwasa sizablehelpinthosedays. Fromindicationsatdepartmentmeetings,IgatheredthatmychancesofstayingonatHarvardonapermanentbasiswerepoor,soIbegantomakeinquiriesabout anotherpositionfor1940.AnassistantprofessorshipbecamevacantatLehighUniversityinBethlehem,Pennsylvania,andIreceivedaletterinvitingmeforan interview.IwasnotmuchinterestedingoingtoBethlehem,butG.D.Birkhofftoldme,"Stan,youmustknowthatitisimpossibleinthiscountrytogetany advancementorraisesinsalarywithoutoffersfromelsewhere.Yes,gohavetheinterviewatLehigh."Ireplied,"Whowillteachmyclassthatday?""Iwill,"hesaid.I feltbothembarrassedandhonoredthatthegreatProfessorBirkhoffwouldcondescendtoteachmyclassinundergraduatemechanics.Indeed,charminglychildishas heoftenwas,andwantingtoshowtheyoungstudentswhohewas,hegaveacomplicatedandadvancedlecturewhich,Ilaterlearned,theydidnotunderstandvery well. BethlehemwascoveredwithayellowpallofacridsmogwhenIarrivedfortheinterview—aninauspiciousbeginning.Thechairmantookmearoundthedepartment andintroducedmetoayoungprofessorthere.IthappenedtobethenumbertheoristD.H.Lehmer.Asweenteredhisoffice,hewascorrectingalargepileofblue booksandsaidtome,rightinfrontofthechairman,"Seewhatwehavetodohere!"Thiscontributedtoanegativeimpressionandbroughtinstantlytomyminda similarsituationinPoland.Whenachambermaidorsomeotherservantgirlwasleavingaplaceofemployment,shewouldtakethenewprospectivemaidasideand showherthelessfavorableaspectsofthejob. ThiswastheperiodofmylifewhenIwasperhapsinthe

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worststate,mentally,nervously,andmaterially.Myworldhadcollapsed.ProspectsforthereconstitutionofPolandinanyrecognizableformweredimindeed.There wasaterribleanxietyaboutthefateofallthosewhomwehadleftbehind—familyandfriends.Adamwasalsoinaverydepressedstate,andthiscontributedtomy worries.L.G.Henderson,whowasalwaysfriendlyandhelpful,gavemeallthemoralsupporthecould.WhenFrancecollapsedinthespringof1940,thesituation becamesodarkandseeminglyhopelessthatdespairgrippedalltheEuropeanémigrésonthissideoftheocean.TherewastheaddedworrythatshouldGermanideas prevailalongwiththeirmilitarysuccesses,lifeinAmericawouldbecomequitedifferent,andxenophobiaandanti­Semitismmightgrowheretoo. DuringthatperiodIlivedinalittleroomonthefourthflooroftheAmbassadorHotel.Onthefifth,theAlfredNorthWhiteheadslivedinalargeapartmentwhosewalls werecuriouslypaintedinblack.IknewWhiteheadfromtheSocietyofFellowsdinners.HeandMrs.Whiteheadheldaweekly"athome"eveningtowhichthey invitedme.Hewasalreadyquiteold,buthismindwascrystalclear,sharp,incisive,withabettermemorythanmanyofhisjuniors.Iremembertheirfortitudeand courageatthetimeofthebombingofLondon.Whiteheadneverseemedtogiveuphopethatthewarwouldbewonintheend,andhelivedtoseethedefeatof Germany. ConversationattheWhiteheads'wasextremelyvaried.Besidesthewar,onediscussedphilosophy,science,literature,andpeople.ThesubjectofBertrandRussell cameuponce.Hewashavinggreattroublesinthiscountry.HehadquarreledwithBarnes,thePhiladelphiamillionairewhoemployedhimatthetime,andhewasin difficultyatCityCollegebecauseofhisviewsonsexandlecturesonfreelove.Harvardtriedtogethim,buttheinvitationdidnotgothroughbecauseofawaveof protestsfromproperBostonians.IrememberWhiteheadtalkingaboutallthisandMrs.Whiteheadexclaiming,"Oh,poorBertie!"Therewasalsotalkofmathematics. Oncesomeoneasked,"Professor

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Whitehead,whichismoreimportant:ideasorthings?""Why,Iwouldsayideasaboutthings,"washisinstantreply. IspentmuchofmytimewiththeotherPoleswhohadfoundtheirwaytoCambridge—Tarski,StefanBergmanandAlexanderWundheiler.Theywereallterribly unhappy,Wundheilermostofall.Healwayshadsomekindof"Weltschmerz."WewouldsitinfrontofmylittleradiowhichIleftonalldaylong,andlistentothewar news.Hewouldstayforhoursinmyroom,andwedrankbrandyfromtoothbrushglasses.Hewasatalentedmathematician,anextremelynice,pleasant,and intelligentperson,withamindratherhardtodescribe—thatofanintelligentcritic,butsomewhatlackingthe"it"ofmathematicalinvention.GeniusisnotthewordI mean.Itishardtodescribethetalentforinnovation,evenonamodestscale;besides,itexistsinacontinuousspectrumandislargelyinfluencedby"luck."Theremay besuchathingashabitualluck.Peoplewhoaresaidtobeluckyatcardsprobablyhavecertainhiddentalentsforthosegamesinwhichskillplaysarole.Itislike hiddenparametersinphysics,thisabilitythatdoesnotsurfaceandthatIliketocall''habitualluck."Itisoftenremarkedthatinsciencetherearepeoplewhohaveso muchluckthatonebeginstofeelitissomethingelse.Wundheilerlackedthisspecialspark. Idon'trememberwhenandhowhefirstappearedintheStates.HehadatemporaryjobatTuftsCollegeinBoston.Hehadtheusualimpressions,complaints, appreciationandadmirationfortheUnitedStates;wetalkedaboutthatatgreatlength.Attitudesofstudentsamusedandshockedhim.Beingusedtothemoreformal Polishmanners,hewasquiteputoutwheninclassonedayastudentcalledouttohim,"Psst!Thewindowisopen.Canyoucloseit?"Onedidnottalktoprofessors thatwayinPoland. HewasinterestedinthegeometryoftheDutchmathematicianSchouten,whichtomewastooformalandsymbolic.Thenotationsweresocomplicated,Imadefunof theformulaesayingthattheyconsideredageometricobjectamere

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symbol,aletteraroundwhichindiceswerehungright,left,up,anddown—likedecorationsonaChristmastree. WegraduallylostcontactafterIleftCambridge.Ilearnedlaterthathehadcommittedsuicide.Ihadapremonitionofthisbecauseofapoemhewouldreciteabouta manwhohangedhimselfwithhistie.Hewaslonely,andmanytimeshehadtoldmeofhisunhappinessbecauseofhislooks.Hewasaveryshortmanwithan intelligentface,butnotonethatwasconsideredappealingtowomen.Hethoughtofhimselfasugly,whichbotheredhim. Inmanycases,mathematicsisanescapefromreality.Themathematicianfindshisownmonasticnicheandhappinessinpursuitsthataredisconnectedfromexternal affairs.Somepracticeitasifusingadrug.Chesssometimesplaysasimilarrole.Intheirunhappinessovertheeventsofthisworld,someimmersethemselvesinakind ofself­sufficiencyinmathematics.(Somehaveengagedinitforthisreasonalone.)Yetonecannotbesurethatthisisthesolereason;forothers,mathematicsiswhat theycandobetterthananythingelse. Towardtheendofthe1940academicyearBirkhoffintimatedtomethattheremightbeavacancyattheUniversityofWisconsin.Headded,"Youshouldnotbelike theotherEuropeanrefugeeswhotryatallcoststostayontheEastCoast.DoasIhavedone,trytogetajobinMadison.Itisagooduniversity;Iwasthereasa youngman."Takinghisadvice,IwenttoameetingoftheAmericanMathematicalSocietyatDartmouthtomeetProfessorMarkIngraham,whowaschairmanofthe mathematicsdepartmentinMadison.Inthosedays,mathematicsmeetingswerejobfairsatwhichonecouldseethedepartmentchairmen—almostlikehereditary caciques,eachsurroundedbysupplicants,groupsofyoungermenlookingforjobs.Thesituationwascompletelyreversedinthelate1950sandearly1960s,whena loneyoungmanfreshfromschool,withabrand­newPh.D.,wouldbesurroundedbychairmenlookingforyoungprofessors.

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AttheDartmouthmeetingIhadacomicaladventure.LateintheeveningIwalkedintomydormitoryroom.Itwasdark,andItriedtogetintobedwithoutputtingthe lighton.AsIsatdown,Iheardasqueakandagroan.Mybedhadanotheroccupant.Igropedfortheotherbed.Theoccupantofmybedsaid,"Dr.Ulam?"I answered,"Yes."Immediatelyhesaid,"Givenagroupwhichissuchandsuch,doesithavethisandthatproperty?"Ithoughtamomentandanswered,''Yes," proceedingtooutlineareason."Ifitiscompact,thenitistrue.""Butifitisn'tcompact,"hetriedtocontinue.Itwaslate,Iwastired,andfeltlikesaying,"Ifitisn't compact,tohellwithit."Ilettheconversationdropandfellasleep. G.D.Birkhoffseemedtolikeandappreciatemywork.IthinkIknowapossiblereason.Helikedmyself­confidenceandmynearimpudenceindefendingthepoint ofviewofmodernmathematicsbasedonsettheoryagainsthismoreclassicalapproach.Headmiredtheso­to­speakhormonal,emotionalsideofmathematical creativity.Iprobablyremindedhimofthewayhefeltwhenhewasyoung.HelikedthewayIgotalmostfuriouswhen—inordertodrawmeout—heattackedhisson Garrett'sresearchongeneralizedalgebrasandmoreformalabstractstudiesofstructures.Idefendeditviolently.Hissmiletoldmethathewaspleasedthattheworth andoriginalityofhisson'sworkwasappreciated. Indiscussingthegeneraljobsituation,hewouldoftenmakeskepticalremarksaboutforeigners.Ithinkhewasafraidthathispositionastheunquestionedleaderof AmericanmathematicswouldbeweakenedbythepresenceofsuchluminariesasHermannWeyl,JacquesHadamard,andothers.Hewasalsoafraidthatthe explosionofrefugeesfromEuropewouldfilltheimportantacademicpositions,atleastontheEasternseaboard.Hewasquotedashavingsaid,"IfAmerican mathematiciansdon'twatchout,theymaybecomehewersofwoodandcarriersofwater."Heneversaidthattome,buthedidoftenmakeslightingre­

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marksabouttheoriginalityofsomeforeigners.Healsomaintainedthattheyoughttobecontentwithmoremodestpositions;objectivelyspeaking,thiswas understandableandevenfair.Justthesame,Iwouldsometimesgetangry.Perhapsbecausemyfamilyhadbeensowelloffandthatuntil1940Ineverhadtogive muchthoughttomyfinancialsituation,IcouldbeindependentandsaidwhatIthoughtopenly.OnceIcounteredoneofhisattacksonforeignerswith,"Whatpleasure doyoufindinplayingagamewheretheoutcomedoesnotdependontheskilloftheopponentbutonsomeexternalcircumstance?Whatpleasureisthereinwinninga gameofchessfromaplayerwhoisforcedtomakepoormovesbecauseheneedshelpfromhisopponent?"Hewasquitetakenbackbythisremark. ButBirkhoffhelpedmetosecurethejobinMadison.HespoketoIngrahamonmybehalf;aftertheDartmouthmeeting,Ireceivedanofferofaninstructorshipin Madison.Atthirty,andwithacertainreputationamongmathematiciansinPolandandAmerica,IfeltIcouldhavebeenofferedatleastanassistantprofessorship.But thecircumstancesweresofarfromnormal,andwithsomeonelikeJacquesHadamard,themostcelebratedFrenchmathematician,havingbeenofferedalectureshipin NewYork,orTarskihavingbeentakenonasaninstructorinBerkeley,Iswallowedmyprideandacceptedtheoffer.Financially,itwasnotbad—somethinglike twenty­threehundreddollarsayear.Nevertheless,IfeltsadatleavingHarvardandthe"culturalEast"forwhatIbelievedtobeamoreprimitiveandintellectually barrenMiddleWest.OntheEastCoastitwasimpliedthatHarvardandperhapsYaleandPrincetonweretheonlyplaceswith"culture."IwassurethatMadison, aboutwhichIknewnothing,wouldbelikeSiberiaandthatIwasbeingexiled.Butsincetherewerenoalternatives,IpreparedtoleaveCambridgeattheendofthe summer,grimlydeterminedtolivethroughtheyearsofexileandtoawaittheoutcomeofthewar.

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Chapter7 TheUniversityofWisconsin 1941–1943 ItraveledtoMadisonbywayofChicago,whereIchangedtoasmallertrainthatmadeseveralwhistlestops,oneatatownnamedHarvard.Theironydidnotescape me,andIfeltfatewasplayingacrueljokeonme.Itdidnottakelongformetochangemyoutlookcompletely.IpromptlyfoundthatthestateofWisconsinhad importantliberalpoliticaltraditions,thatthefamousLaFolletteshadlefttheirimprintnotonlyonthestatecapitalbutontheUniversityaswell.Theentirephysical impression,thelandscape,thelakes,thewoods,thehouses,andthesizeofthecityweremostagreeable.Livingconditionswereapleasantsurprise.Iwasgivena roomattheUniversityClubandalmostatoncemettherecongenial,intelligentpeoplenotonlyinmathematicsandscience,butalsointhehumanitiesandarts.The roomsweresmall,withbath,bed,desk,andchairs.(IrememberedhowinoneofAnatoleFrance'snov­

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els,thehero,FatherCoignard,says,"Alloneneedsisatableandabed.Atableonwhichtohaveinturnlearnedbooksanddeliciousmeals,abedforsweetrepose andferociouslove!")Downstairstherewerecomfortablecommonrooms,alibrary,diningroom,evenagameroomwithbilliardtables. Theuniversitywasadequatelysupportedbystatefundsandhadextraincomefromaformerprofessor'sdiscoveryofaspecialtreatmentofmilk,thepatentrightsfor whichbelongedtotheUniversity. Johnny'sfriendEugeneWignerwasaphysicsprofessorthere.Ihadaletterofintroductiontoanothereminentphysicist,GregoryBreit,fromHarlowShapley,the HarvardastronomerwithwhomIhadhadpleasantscientificandsocialcontactsduringmystayinCambridge.ItwasShapleywhodiscoveredthe"scaleofthe universe"byusingtheluminosityperiodofthecepheidstarsasmarkers.Iquicklymadefriendswithmostofthemathematicians—manyofthemmycontemporaries, SteveKleenethelogician,C.J.Everett,DonaldHyers,andothers.Beingbynaturegregarious,IlikedlivingattheUniversityClub,meetingandtakingmealswith interestingcolleagues. OneofthemwasVassilief,aRussianémigré,agreatexpertonByzantinehistory,andalmostacharacteroutofNabokov'sbookPnin.Atdinner,healwaysordered asecondbowlofsoupandwouldsaytome,"Americansarefunny;evenwhenthesoupisexcellent,theyneverthinkoforderingasecondbowl."Likemany Russians,helikedtodrinkandcarriedasmallflaskofvodkainhiscoatpocket.Hemusthavebeeninhissixtiesatthetime.Sometwoyearslater,whentheU.S. ArmytookovertheFacultyClubforitsquarters,Vassiliefandtheotheroccupantshadtofindotherhousing.Vassiliefwasgivenatwo­roomsuiteinaprivatehouse. Hewasenchantedwiththisnewspaciousness."It'swonderful,"heexplained.''Youcansleepinoneroomandworkintheother."AndjustlikePninhethrewwhathe called"ahouse­heatingpartytocelebrate."

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AnotherinterestingpersonwasaprofessorofEnglishliterature,abachelor,ProfessorHanley.Thankstomymemory,whichenabledmetoquoteLatinandtodiscuss GreekandRomancivilization,itbecameobvioustosomeofmycolleaguesinotherfieldsthatIwasinterestedinthingsoutsidemathematics.Thisledquicklytovery pleasantrelationships.Hanleywasagoodbilliardplayer.Heinsistedonteachingmehowtoplay,thoughhewasratherappalledbymyineptitude.This,Ifound,isa veryniceandveryAmericantrait—thedesiretocoachandinstruct. SoIfoundMadisonnotatalltheintellectualdesertIhadfeareditwouldbe.Theuniversityhadatraditionofexcellenceinseveralfieldsofthenaturalsciences.Ithad greatexpertiseinlimnology.Limnology,thescienceoflakes,wasdevelopedbyanoldprofessorwhosenameIcannotrecall,butwhousedtosay,Iwastold,that everytimeherememberedthenameofastudentheforgotthenameofafish.Biology,too,wasstrongattheUniversityofWisconsin,aswellaseconomicsand politicalscience.TheeconomistSeligPerlman,andNathanFeinsinger,wholaterbecamenationallyknownasalaborrelationsexpert,werethereatthetimeamong manyothereminentprofessors. Italsoseemedthatforeignerslikemyselfwhowere,sotospeak,notunpresentable,werewellreceivedintotheacademiccommunity'ssociallifeandquickly establishedgoodrelationswithmanyprofessorsinvariousfields.Theintellectualatmospherewaslively.Onthewholetheprofessorsdidnotputonairsasafewhave atHarvard.Onthecontrary,perhapsinordertobearcomparisonwiththefamousolderuniversities,theyworkedmoreenergetically,butthe"Oh,excusemeI'vegot togettowork"syndromewasnotasevidentasatHarvard. SomethingelsehappenedtomakeMadisonmostimportanttome.ItwastherethatImarriedaFrenchgirl,whowasanexchangestudentatMountHolyokeCollege andwhomIhadmetinCambridge,FrançoiseAron.Marriage,ofcourse,changedmywayoflife,greatlyinfluencingmydaily

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modeofwork,myoutlookontheworld,andmyplansforthefuture. ThepoetWilliamElleryLeonard,atallmanwithaverylargeheadandamaneofwhitehair,wasoneoftheinterestingandcolorfulmembersofthefaculty,agreat eccentric.TheauthorofthebookTheLocomotiveGod,hewasreputedtohaveanintense,neuroticfearoftrains.Thispreventedhimfromeverleavingthe UniversityinMadison;thestorywasthathissalary(whichwasverylowforafullprofessor)neverwasincreased,becausehewouldneverleaveanyway.Ifoundthis reasonratherhumorous. Atthattime,atmanyuniversitiesdeansandchairmenoftenrantheirdepartmentsnotsomuchforexcellenceinscholarlyoreducationalpursuits,butforgoodeconomy andefficiency—ratherlikeabusiness.Indeed,shortlyafterIarrived,someonepointedoutthatthemarvelousphysicallocationofthecampusontheshoresofLake Mendotaconstitutedapartofoursalaries,makingthemalittlelowerthanatothercomparablestateuniversities.Thismademejokewithsomeofmyyounger colleagues:everytimewelookedatthebeautifullakeitwascostingusabouttwodollars.AtoneofthefirstfacultymeetingsIattended,ClarenceA.Dykstra,the President,veryimposinginappearance(andactuallyaverygoodman),startedhisspeechwith,"Allofusfaceachallengethisyear."AtthisInudgedmyneighborand whispered,"Watchout!Thismeansnoraisesforthefaculty.Sureenough,tenminuteslaterDykstrasaidsomethingtothiseffect,andmyneighborlaughedoutloud. TheastronomerJoelStebbinswasaprofessorinMadison.Ienjoyedmeetinghimandtalkingtohimintheobservatory.Hehadanexcellentsenseofhumorandwasa greatpracticaljoker.Onceonaclear,coldsunnywinterSunday,hedrovetoourapartmentandhonkedthecarhorn.Ilookedoutandtherehewas,saying,"Would youliketocomewithmetotheYerkesObservatory?ThereisameetingoftheAstronomicalSocietythere."YerkeswasnotfarfromMadison—aboutatwo­hour drive.Idressedwarmly,hurried

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downandonthewaywediscussedallkindsofproblems.Thenteasinglyhesaid:"Wouldyouliketogiveatalk?"Toanswerajokewithajoke,Iresponded,"Yes, forfiveortenminutes."QuicklyIstartedtothinkaboutwhatIcouldsaytoastronomersinafewminutes.IrememberedthatonceIhadbeenthinkingaboutthe mathematicsofthewaytrajectoriesofcelestialbodiesmightlookfromamovingsystemofcoordinatesandhow,byasuitablemotionoftheobserver,onecouldmake complicated­lookingorbitsappearsimplerifoneassumedthattheobserverwashimselfmoving.Icalledthisgeneralquestion''TheCopernicanProblem"andspoke forafewminutesonthat.Indeed,itisaworthwhilesubjecttoconsider,anditdoesgiverisetosomebonafidetopologicalandmetricquestionsinwhichIhad obtainedafewsimpleresults. FromthefirstyearofmyemploymentIwasgivenaverylightteachingload—namelyonlyelevenhoursofelementarycourses(recognizingthefactthatIwasdoing researchandwritinganumberofpapers),whilesomeoftheotherinstructorstaughtthirteenorsixteen.Later,thiswasreducedtoninehoursperweek.These elementarycoursesrequiredessentiallynopreparationonmypart,asidefromanoccasionallookatthesequenceoftopicsinthebooksothatIwouldcoverthe prescribedmaterialandnotgotooquicklyortooslowly.Buttheveryexpression"teachingload"asusedbyalmosteveryonefromfamousscholarstoadministrators wasnotonlyrepugnanttomebutridiculous.Itimpliedphysicaleffortandfatigue—twothingsIhavealwaysbeenafraidof,lesttheyinterferewithmyownthinking andresearch.IwasgratefultoIngraham,theheadofthedepartment,forunderstandingthis.Hewasajovialandpleasantman,whousedtocometothefacultyclub onweekendstowatchmoviesoffootballgames.Knownforhisfondnessforapplepiewithcheese,heintroducedmetothecheesesofWisconsin,oneofthestate's dairyspecialties,beforeImadealateracquaintancewiththeinfinitevarietyofthoseofFrance.

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Ingeneral,teachingmathematicsisdifferentfromteachingothersubjects.Ifeel,asdomostmathematicians,thatonecanteachmathematicswithoutmuchpreparation, sinceitisasubjectinwhichonethingleadstoanotheralmostinevitably.Inmyownlecturestomoreadvancedaudiences,seminars,andsocieties,Idiscusstopicsthat currentlypreoccupyme;thisismoreofastreamofconsciousnessapproach. IamtoldthatIteachratherwell.ThisispossiblybecauseIbelieveoneshouldconcentrateontheessenceofthesubjectandalsonotteachallitemsatauniformlevel. Iliketostresstheimportantand,forcontrast,afewunimportantdetails.Oneremembersaproofbyrecalling,asitwere,asequenceofpleasantandunpleasant points—thatis,difficultandeasyones.Firstcomesadifficultyonwhichonemakesaneffort,thenthingsgoautomaticallyforawhile,suddenlyagainthereisanew specialtrickthatonehastoremember.Itissomewhatlikegoingthroughalabyrinthandtryingtoremembertheturns. WhenIwasteachingcalculusinMadison(anditisamarvelousthingtoteach)andhadworkedoutaproblemontheboard,Iwasamusedwhensomestudentwould say,"Doanotheronelikeit!"Theydidn'tevenhaveanamefor"it."Needlesstosay,thesestudentsdidnotbecomeprofessionalmathematicians. Onemaywonderwhetherteachingmathematicsreallymakesmuchsense.Ifonehastoexplainthingsrepeatedlytosomebodyandassisthimconstantly,chancesare heisnotcutouttodomuchinmathematics.Ontheotherhand,ifastudentisgood,hedoesnotreallyneedateacherexceptasamodelandperhapstoinfluencehis tastes.Apriori,Itendtobepessimisticaboutstudents,eventhebrightones(thoughIremembersomegoodstudentsatHarvardwithwhomIcouldtalkandfeelthat teachingwasnotmerelyanemptyexercise). Generallyspeaking,Idon'tmindteachingassuch,althoughIdonotliketospendtoomuchtimeatit.WhatIdislikeistheobligationtobeatagivenplaceatagiven

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time—notbeingabletofeelcompletelyfree.Thisisbecauseoneofmycharacteristicsisaspecialkindofimpatience.WhenIhaveafixeddate,evenapleasantdinner orparty,Ifret.AndyetwhenIamcompletelyfree,Imaybecomerestless,notknowingwhattodo. WithmyfriendGian­CarloRotaIoncecomputedthatincludingseminarsandtalkstoadvancedaudiences,wemusthavelecturedforseveralthousandhoursinour lives.Ifonerecognizesthattheaverageworkingyearinthiscountryamountstoaroundtwothousandhours,thisisalargeproportionofone'swakingtime,evenifitis notcompletely"waking"sinceteachingissometimesdoneinapartialtrancelikestate. ItwasinMadisonthatImetC.J.Everett,whowastobecomemyclosecollaboratorandgoodfriend.EverettandIhititoffimmediately.Asayoungmanhewas alreadyeccentric,original,withanexquisitesenseofhumor,wry,concise,andcausticinhisobservations.Hewastotallydevotedtomathematics—theywerehisonly interest.IfoundinhimmuchthatresembledmyfriendMazurinPoland,thesamekindofepigrammaticcommentsandjokes.Physically,theyhadacertainsimilarity, bothbeingthin,bony,andlessthanmediumheight.Eventheirhandwritingwassimilar;theybothwroteinveryneat,almostmicroscopiclittlesymbols.Everettis severalyearsyoungerthanI.Wecollaboratedondifficultproblemsof"order"—theideaoforderforelementsinagroup.Inourmathematicalconversations,as always,Iwastheoptimist,andhadsomegeneral,sometimesonlyvagueideas.Hesuppliedtherigor,theingenuitiesinthedetailsoftheproof,andthefinal constructions. Onepaperwewroteonorderedgroupscaughtthefancyofthewomanwhowasheadofoneofthewomen'smilitaryorganizationsduringthewar.Atameetingwe heardherdescribetheactivitiesofthecorpsbycallingtheorganization"orderedgroups." Laterwewroteanotherjointpaperonprojectivealgebras.Ithinkthiswasthefirstattempttoalgebraizemathe­

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maticallogicbeyondtheso­calledBooleanorAristotelianelementaryoperationstoincludetheoperations"thereexists"and"forall,"whicharebothvitaland comprehensiveforadvancedmathematics. Webothtaughtcoursesfornavalrecruitsin1942and1943.Also,inordertoearnsomeextramoney,wecorrectedpapersfortheArmyCorrespondenceSchool. Françoisehelpedinthat,too—shecoulddoitverywellforelementaryarithmeticandalgebraexercises.TheCorrespondenceSchoolpaidthirty­fivecentsperpaper; thisamountedtoquiteabitofmoneyandbegantoreachsumscomparabletotheuniversitysalary.Atthispoint,theadministrationdecidedtostepinandimpose restrictionsonthenumberofpapersonepersoncouldcorrect.TheArmycorrespondenceworkwasadministeredbyanolderwomanwhowasamemberofthe mathematicsdepartment;itwassupervisedbyaprofessor,HerbertEvans,averyjovialandpleasantpersonwithwhomIbecamefriendly.Hewasoneofthemost good­humoredpersonsIhaveeverknownanywhere. EverettandIsharedanofficeinNorthHall,anoldstructurehalfwayupthehillwhichhousedthemathematicsdepartment.LeonCohen,avisitingprofessorfromthe UniversityofKentuckywithwhomwehadpublishedsomejointwork,wastherewithus.Wespenthoursinthatoffice,thethreeofus;theentirebuildingwould resoundwithourfrequentlaughter.Beforeandafterclasses,wecorrectedstudentpapers—anoccupationIhatedandalwaystriedtoputoff.Asaresultmydesk waspiledhighwithstacksofuncorrectedworkbooksandasIdepositednewonesatoneend,theolderonesattheothermercifullydroppedintothewastebasket. SometimesthepoorstudentswonderedwhyIwasnotreturningtheirwork. Afterlunch,weplayedbilliards—ortriedto.Hanley'slessonsattheFacultyClubhadverylittleeffectonmygame.FuninthisNorthHallofficeandourfrequent sessionsintheStudentUnion,aluxuriousbuildingontheshoreofthelake,wereamongthecharmsoflifeinMad­

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ison.Thiscombinationofleisureplusinformalstimulationplaysanimportantroleinone'smentalactivity.Beyondthemerelyagreeablephysicalsetting,itisoftenmore valuablethanthemoreformalgatheringsatseminarsandmeetingswhichleadtodiscussionsofadriertype.Initsway,thissomewhatreplacedformetheoldsessions inthecoffeehousesofLwówforwhichIhavehadalongingeversinceleavingPoland. EverettstayedinMadisonthroughoutthewar.LaterhejoinedmeatLosAlamos,wherewedidmuchmoreworktogetherincludingournowwell­known collaborationandworkontheH­bomb. Everettexhibitedatraitofmindwhoseeffectsare,sotospeak,non­additive:persistenceinthinking.Thinkingcontinuouslyoralmostcontinuouslyforanhour,isat leastforme—andIthinkformanymathematicians—moreeffectivethandoingitintwohalf­hourperiods.Itislikeclimbingaslipperyslope.Ifonestops,onetendsto slideback.BothEverettandErdöshavethischaracteristicoflong­distancestamina. TherewerealsoDonaldHyersandDorothyBernstein.Hyersalsohadpersistenceinthinkingaboutproblemsandanabilitytocontinuetopushthetrainofthoughton aspecificproblem;wewroteseveraljointpaperstogether.DorothyBernsteinwasagraduatestudentinmyclass.Shewasaninterested,enthusiastic,andfaithful takerofnotesandorganizerofmaterialfromacourseItaughtonmeasuretheory.Shecollectedmuchmaterial,andweintendedtowriteabooktogether,butour workwasinterruptedbymydeparturefromMadisonin1944andourplanswereabandoned. Onedayinmyoffice,ayoungandbrilliantgraduatestudentnamedRichardBellmanappearedandexpressedadesiretoworkwithme.Wediscussednotonly mathematicsbutthemethodologyofscience.WhentheUnitedStatesenteredthewar,hewantedtogobackEast—IbelievetoNewYorkwherehecamefrom— andaskedmetohelphimobtain

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afellowshiporastipendsohecouldcontinueworkingafterheleftMadison.IrememberedthatinPrincetonLefschetzhadsomenewscientifico­technological enterpriseconnectedwiththewareffort.IwrotetohimaboutBellmaninasortofMachiavellianway,sayingthatIhadaveryablestudentwhowassogoodthathe deservedconsiderablefinancialsupport.IaddedthatIdoubtedthatPrincetoncouldaffordit.This,ofcourse,immediatelychallengedLefschetzandheoffered Bellmanaposition.Twoyearslater,DickBellmanappearedsuddenlyinLosAlamosinuniformasamemberoftheSEDs,aspecialengineeringdetachmentofbright andscientificallytalenteddrafteeswhohadbeenassignedtohelpwiththetechnicalwork. ThroughmyconnectionswithphysicistsandaseminarItaughtinphysicsintheabsenceofGregoryBreit,IheardoftherecentarrivalinMadisonofaveryfamous Frenchphysicist,LéonBrillouin.Icalledonhimandfoundthathiswife,Stéfa,wasPolish,borninthecityofLódz,alargetextile­manufacturingtown.StéfaandLéon hadmetwhenshewasayoungstudentinParis,andtheyweremarriedsometimebeforeWorldWarI.WhenWorldWarIIbrokeouthewas,Ithink,adirectorof theFrenchbroadcastingservice,withallthemilitaryresponsibilitiesthatentailed.AfterthecollapseofFranceandtheinstallationoftheVichyrégime,hemanagedto escapefromFranceatthefirstopportunity.Hewasinternationallyknownforhisworkinquantumtheory,statisticalmechanics,andalsoinsolidstatephysics.Infact, hewasoneofthepioneersinthetheoryofsolidstate.(Theideaof"Brillouin"zonesandotherimportantnotionsareduetohim.)Hewasalsoaveryprolificand successfulwriterofphysicstextbooksandmonographs. Mrs.Brillouinhadagreatartisticflair.Intheearlynineteen­twentiessheacquired,formodestsums,manyworksbyModigliani,Utrillo,Vlaminck,andotherpainters. InMadisonsheherselfstartedtopaintflowercompositionsinoilmadetipofthicklayers—astyleallherown.TheBrillouinsinvitedustostopattheirapartmentthe day

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FrançoiseandIweremarried.Theyheldasmallsurprisereceptionforus,wherewedrankFrenchchampagneandpartookofamemorablecakebyStéfa.Stéfa BrillouinspokehardlyanyEnglish,butafewweeksafterherarrival,inshoppingforvariousobjectsshediscoveredthat"lecentimètred'ici"asshecalledourinch,was abouttwoandahalftimesthe"centimètredeFrance."Thisalmostexactestimate,theinchbeing2.54centimeters,wasobtainedsolelybylookingatthesizesof materials,curtains,andrugs.InMadisonacloseassociationbegan,whichcontinuedlongafterthewarandlasteduntiltheirdeathsafewyearsago. BeforemysecondyearinMadison,Iwaspromotedtoanassistantprofessorship—astepwhichgavemehopeandsomeconfidenceinthematerialaspectsofthe future.Tostartahomeandatthesametimehelpsupportmybrotheronmymodestsalary(oftwenty­sixhundreddollarsayear)wasdifficult.Oftentomakeends meetIvisitedtheFacultyCreditUnion,whereasympatheticofficermademeloansofuptoonehundreddollars,whichhadtoberepaidinafewmonths'time. Iwasaskedtorunthemathematicscolloquium,whichtookplaceeverytwoweeksandinvolvedbothlocalandvisitingmathematicians.Imightaddthatthepayments tospeakerswereridiculouslysmall;evenforthosetimes,theyamountedtoabouttwenty­fivedollars—andthisincludedtravelingexpenses. ThecolloquiumwasrundifferentlyfromwhatIhadknowninPoland,wherespeakersgaveten­ortwenty­minuteinformaltalks.AtMadisontheywereone­hour lectures.ThereisquiteadifferencebetweenshortseminartalkslikethoseatourmathsocietyinLwów,andthetypeoflecturewhichnecessitatestalkingaboutmajor efforts.Thelatterwerebetterprepared,ofcourse,buttheirgreaterformalityremovedsomeofthespontaneityandstimulationoftheshorterexchanges.Inthis connectionImetAndréWeil,thetalentedFrenchmathematician,whohadgonetoSouthAmericaatthebeginningofthewar.Hedislikedconditions

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thereandcametotheUnitedStates,wherehehadgottenajobatLehigh.Weilwasalreadywellknowninternationallyforhisimportantworkinalgebraicgeometry andingeneralalgebra.HiscolloquitimtalkwasononeofhismostimportantresultsontheRiemanhypothesisforfieldsoffinitecharacteristic.TheRiemanhypothesis isastatementthatisnoteasytoexplaintolaymen.Itisimportantbecauseofitsnumerousapplicationsinnumbertheory.Ithaschallengedmanyofthegreatest mathematiciansforaboutahundredyears.Itisstillunprovedalthoughconsiderableprogresshasbeenmadetowardapossiblesolution. DeanMontgomery,afriendIhadmetatHarvard,cameatmyinvitationandgaveacolloquiumtalk.Therewasavacancyinthedepartment,andItriedtointeresthim incomingtoouruniversity,whereIngrahamandLanger,thetwomostseniorprofessors,werebothverymuchinfavorofhisappointment;insteadhewenttoYale. LaterhetoldmestoriesabouttheatmosphereatYale,whichatthattimeinsomecircleswasultra­conservative.Inhisinterviewhewasaskedwhetherhewasforor againstJewsintheacademicprofessionandalsowhetherhewasaliberal.Eventhoughheansweredbothquestions"wrongly"fronttheinterlocutor'spointofview,he neverthelessreceivedanoffer.HeleftYaleafewyearslatertojointheInstituteinPrinceton. EilenbergandErdöswerealsoamongthespeakersinvitedtothecolloquia.ErdöswasoneofthefewmathematiciansyoungerthanIatthatstageofmylife.Hehad beenatruechildprodigy,publishinghisfirstresultsattheageofeighteeninnumbertheoryandincombinatorialanalysis. BeingJewishhehadtoleaveHungary,andasitturnedout,thissavedhislife.In1941hewastwenty­sevenyearsold,homesick,unhappy,andconstantlyworried aboutthefateofhismotherwhohadremainedinHungary. HisvisittoMadisonbecamethebeginningofourlong,intense—albeitintermittent—friendship.Beinghard­upfinancially—"poor,"asheusedtosay—hetendedto extend

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hisvisitstothelimitsofwelcome.By1943,hehadafellowshipatPurdueandwasnolongerentirelypenniless—"evenoutofdebts,"ashecalledit.Duringthisvisit andasubsequentone,wedidanenormousamountofworktogether—ourmathematicaldiscussionsbeinginterruptedonlybyreadingnewspapersandlisteningto radioaccountsofthewarorpoliticalanalyses.BeforegoingtoPurdue,hewasattheInstituteinPrincetonformorethanayearwithonlyapittanceforsupportthat wastoendlater. Erdösissomewhatbelowmediumheight,anextremelynervousandagitatedperson.Atthattimehewasevenmoreinperpetualmotionthannow—almostconstantly jumpingupanddownorflappinghisarms.Hiseyesindicatedhewasalwaysthinkingaboutmathematics,aprocessinterruptedonlybyhisratherpessimistic statementsonworldaffairs,politics,orhumanaffairsingeneral,whichhevieweddarkly.Ifsomeamusingthoughtoccurredtohim,hewouldjumpup,flaphishands, andsitdownagain.Intheintensityofhisdevotiontomathematicsandconstantthinkingaboutproblems,hewaslikesomeofmyPolishfriends—ifpossible,even moreso.Hispeculiaritiesaresonumerousitisimpossibletodescribethemall.Oneofthemwas(andstillis)hisownverypeculiarlanguage.Suchexpressionsas "epsilon"meaningachild,"slave"and"boss"forhusbandandwife,"capture"formarriage,"preaching''forlecturing,andahostofothersarenowwellknown throughoutthemathematicalworld.Ofalltheresultsweobtainedjointly,manyhavestillnotbeenpublishedtothisday. Erdöshasnotchangedmuchastheyearshavegoneby.Heisstillcompletelyabsorbedbymathematicsandmathematicians.Nowoversixty,hehasmorethanseven hundredpaperstohiscredit.Amongthemanysayingsabouthimonegoes,"Youarenotarealmathematicianifyoudon'tknowPaulErdös."Thereisalsothewell­ knownErdösnumber—thenumberofstepsittakesanymathematiciantoconnectwithErdösinachainofcollaborators."Numbertwo,"forex­

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ample,istohaveajointpaperwithsomeonewhohaswrittenapaperwithErdös.Mostmathematicianscanusuallyfindalinkwithhim,ifnotinonethenintwo stages. Erdöscontinuestowriteshorthandwrittenlettersbeginning,"Supposethatxisthusandso..."or"SupposeIhaveasequenceofnumbers..."Towardtheend,he addsafewpersonalremarks,mainlyaboutgettingold(thisstartedwhenhewasthirty)orwithhypochondriacorpessimisticobservationsaboutthefateofouraging friends.Hislettersareneverthelesscharmingandalwayscontainnewmathematicalinformation.OurcorrespondencestartedbeforetheMadisonyears,containing manydiscussionsofthehardshipsofyoungmathematicianswhocouldnotfindjobsorofhowtodealwithofficialsandadministrators.Heusedtheexpression,"Oh, heisabigshot"aboutyoungAmericanassistantprofessors,andwhenhecalledmeone,Iintroducedasubclassificationof"bigshots,smallshots,bigfryandsmall fry''—fourordersofstatus.In1941,asanassistantprofessorItoldhimIwasatbesta"smallshot."Thisamusedhim,andhewouldallotoneofthesefourgradesto ourfriendsinconversationsorcorrespondence. Throughthickandthin,JohnnyvonNeumannandIalsocontinuedourcorrespondence,whichincludedalittlemathematicseveninthosedays,andalotaboutthe tragichappeningsintheworld.TherewasmuchisolationismintheUnitedStates,andtheobviousandwidespreaddisinclinationtoenterthewarcreatedinmea feelingofdespairmixedwithresentment.Onthewhole,JohnnywasmoreoptimisticandknewbetterthanIthepoweroftheUnitedStatesandthelong­rangegoalsof U.S.policies.HewasalreadyanAmericancitizenengaged(thoughIdidnotknowitatthetime)inthewareffortthecountrywaspreparing. Thetoneofourmathematicalcorrespondenceandofourconversationswhenwemetatmathematicsmeetingschangedfromtheabstracttomoreapplied,physics­ relatedtopics.Hewasnowwritingaboutproblemsofturbulenceinhydrodynamics,aerodynamics,shocks,andexplosives.

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Johnnyhelddiscussionswithmanyscientists,amongthemNorbertWiener.AlthoughNorbertwasapacifist,hebadlywantedtocontributesomethingimportanttothe Americanwareffort.WienerfeltasRusselldidthatthiswasa"justwar,"anecessarywar,andtheonlyhopeformankindlayinU.S.interventionandvictory.But Norbertwasdifficultinhisdealingswiththemilitary,whereasJohnnyalwaysgotalongwiththem. WienerwroteinhisautobiographythathehadideassimilartotheonesIlaterproposedastheMonteCarlomethod.Hesaysvaguelythathefoundnoresponsewhen hetalkedtosomeoneandsodroppedthematter,inthesamewaythathelostinterestintheideaofgeometryofvectorspacesandfunctionspacesàlaBanach.In fact,inoneofhisbookshecalledthesevectorspaces(whichareassociatedwithBanach'snamealone)Banach­Wienerspaces.Thisnomenclaturedidnot"take"at all. InthefirstWorldWarmathematicianshaddonemuchworkinclassicalmechanics,calculationsoftrajectories,andexternalandinternalballistics.Thisworkwas resumedatthebeginningofthesecondWorldWar,althoughitsoonturnedoutnottobethemainthrustofthescientificapplications.Hydrodynamicandaerodynamic questionsbecamemoredetailedandurgent,particularlybecausetheyweredirectlyconnectedwithspecialwarproblems.Earlyin1940Itookfromthelibrarya Germantextbookonballisticsandstudiedit,butnoticedthattherewasnotmuchinitofimportancetothemilitarytechnologyoftheforties.Atthebeginningofthe warelectroniccomputingmachinesdidnotexist.TherewereonlythebeginningsofthemechanicalrelaymachinesconstructedatHarvard,atIBM,andoneortwo otherplaces. AssoonastheprescribedtimehadelapsedIappliedforandreceivedAmericancitizenshipinMadisonin1941.Ihopedthatthiswouldmakeiteasierformetoenroll inthewareffort.TopasstheexaminationIstudiedthehistoryoftheUnitedStates,theessenceoftheConstitution,the

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namesofPresidents,andtheothertopicsonewaslikelytobeaskedaboutattheexamination.Idon'trememberwhy,butinsteadofmyhavingtogotoChicago,an examinercametoMadisontoourapartment.AfterafewwordsInoticedthathemusthavebeenanimmigranthimselforthesonofanimmigrant.Hisappearancewas quiteJewish,andperhapsimpudentlyIaskedhimabouthisownoriginsandbackground.Hedidnotseemtomindandrepliedthathisparentshadcomefromthe Ukraine.SoonIrealizedwithembarrassmentthatitwasIwhowasexamininghim. ImmediatelyafterIreceivedthecitizenshippapers,ItriedtovolunteerintheAirForce.AtthirtyIwasconsideredtoooldtobecomeacombatpilot,butwithmy mathematicalbackgroundIthoughtIcouldreceivetrainingasanavigator,becausetheUniversityhadreceivedanoticethattheAirForcewaslookingforvolunteers. IwenttoarecruitingcenternotfarfromMadisonforaphysicalexamination.ItwasgivenbyWestCoastJapanesemedicswhohadbeenrelocatedinthisMiddle Westerncamp.Becausethephysicaltestsinvolvedthetakingofbloodsamples,ItoldmyselfjokinglythatIwaslosingsomebloodtotheJapaneseindefenseofmy newcountry.Iwasdisappointedwhenmyapplicationwasrejectedbecauseofmypeculiareyesight. Teachingarmymathcoursesdidnotseemsufficientlyrelevant;Iwantedtodosomethingmoreimmediatelyuseful,somethingthatwouldcontributemoredirectly.I thoughtofgoingtoCanadatoenlistthere,andIrememberedaconversationinCambridgein1940withWhitehead,whohadrelativeswhowereofficersintheRoyal CanadianAirForce.SoIwrotehimaskingwhetherhecouldhelpmetobecomeinvolvedinthewareffortinCanada.HesentbackaletterwhichItreasureforallthe thingsitsaid.EventhoughhesaidthathehadwrittentosomeoneinCanadaonmybehalf,nothingcameofthis. ThenLosAlamosenteredthepicture.

S.M.Ulam,1938.PencilsketchbyZygmundMenkès

StefanBanach,Poland,ca1945

Polishmathematicsstudentsconvention,Lwów,1930 Designatedbynumbers:(1)LeonChwistek,(2)StefanBanach,(3)Stanislaw Loria,(4)KazimirKuratowski,(5)StefanKaczmarz,(6)JuliuszSchauder,

(7)MarceliStark,(8)KarolBorsuk,(9)EdwardMarczewski, (10)S.M.Ulam,(11)A.Zawadzki,(12)EdwardOtto,(13)W.Zonn, (14)M.Puchalik,(15)K.Szpunar

OnthedockatGdyniabeforeStanandhisyoungerbrother,Adam,embarked forAmericain1939.ThefourUlamsare(lefttoright)Jozef(father),Adam, Szymon(uncle),andStan.

JohnvonNeumann,Princeton,1932

TheHarvardSocietyofFellows,Cambridge,1938 Lefttoright,seated:GeorgeHomans,JimFisk,PaulSamuelson,John Snyder,JamesMiller,IvanGetting,WillardQuine,RobertWoodward, GeorgeHass Standing,firstrow:JamesBaker,KennethMurdock,PaulWard,George Haskins,L.J.Henderson,JohnFerry,GeorgeHanfmann,CharlesCurtiss, AlfredNorthWhitehead,JohnLivingstonLowes,TalbotWaterman,Tom

Chambers,SamuelEliotMorison,JohnMiller,ConradArensberg,David Griggs,WilliamWhyte Backrow:F.EdwardCranz,ReedRollins,HarryLevin,Frederick Watkins,JohnOxtoby,E.BrightWilson,RichardHoward,AlbertLord, GarrettBirkhoff,CraigLaDrière,StanUlam,OrvilleBailey (HarvardUniversity)

UlamandC.J.EverettinfrontofNorthHallattheUniversityofWisconsin, Madison,1941

EnricoFermiinthe1940s(HaroldAgnew)

NobelprizewinnersErnestO.Lawrence,Fermi,andI.I. Rabi,duringameetingattheLodge,LosAlamos,inthelate 1940s

LunchattheLodge;(clockwisefromright)RichardFeynman, CarsonMark,JackClark,Fermi

VonNeumann,Feynman,andUlamontheporchofBandelierLodgein FrijolesCanyon,NewMexico,duringapicnic,ca1949(NicholasMetropolis)

TennisatLosAlamos,1958;(lefttoright)JamesTuck,Ulam,Conrad Longmire,DonaldDodder(LosAlamosScientificLaboratory)

ClaireUlam,agedone, withherfather,1945

UlamstandingonthepieceoflandinSantaFethatheboughtfor$150in 1947

ThevonNeumannsstartingthedescentintotheGrandCanyononanexcursion inthelate1940s:Klari,withvisor,isfourthfromfront;Johnny,bareheaded andincitysuit,islast,ontheonlymulefacingthewrongway

OnthePlazainSantaFe,ca1949,vonNeumannandhiseleven­year­old daughter,Marina

Johnny,Claire,andStanintheUlamyardatLosAlamos,ca1954

GeorgeGamow'srepresentationofthe''super"directingcommittee;(bottom; lefttoright):Ulam,EdwardTeller,Gamow;JosephStalinand J.RobertOppenheimerobserveproceedingsfromabove(TheVikingPress)

"Demonstrating"theMANIACtoClaireatalaboratoryopen house,1955(LosAlamosScientificLaboratory)

UlamfoiledbythetechnologicalproblemoffittingTheodore vonKármanwiththeneckcordofamicrophone(MaxSpring, LosAlamosScientificLaboratory)

Aninformalmomentduringapresidentialvisit,LosAlamos,1962;(left toright)VicePresidentLyndonB.Johnson,PresidentJohnF.Kennedy, SenatorClintonP.Anderson,mathematicianUlam,CongressmanJoseph M.Montoya,scienceadvisorJerryWiesner(LosAlamosScientificLaboratory)

UlamandLifeMagazinephotographerAlfredEisenstaedtneartheLos Alamoslaboratory,1962(WilliamH.Regan,LosAlamosScientificLaboratory)

StanandFrançoiseUlamathomeinLosAlamos,1964(LloydShearer)

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PARTIII LIFEAMONGTHEPHYSICISTS

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Chapter8 LosAlamos 1943–1945 Duringthelatespringof1943,IwrotetovonNeumannaboutthepossibilityofwarwork.Iknewhewasinvolved,becausehislettersoftencamefromWashington ratherthanfromPrinceton.Iwasnothappywithteaching,althoughIdidalotofmathematics,wrotepapers,organizedcolloquia,andtaughtwar­relatedcourses.Still itseemedawasteofmytime;IfeltIcoulddomoreforthewareffort. OnedayJohnnyansweredwithanintimationthattherewasinterestingworkgoingon—hecouldnottellmewhere.FromPrincetonhesaidhewasgoingwestvia Chicago,andsuggestedthatIcometotheUnionStationtheretotalktohimsincehehadtwohoursbetweentrains.Thatwasintheearlyfallof1943. Iwentand,sureenough,Johnnyappeared.Whatcaughtmyattentionwerethetwomenescortinghim,lookingabitlike"gorillas."Theywereobviouslyguards,and thatim­

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pressedme;hemustbeanimportantfiguretoratethis,Idecided.Oneofthemenwenttodosomethingabouthisrailroadticket,andwetalkedinthemeantime. JohnnysaidthattherewassomeveryexcitingworkgoingoninwhichIcouldpossiblybeofgooduse;hestillcouldnottellmewhereitwastakingplace,buthe traveledratheroftenfromPrincetontothatlocation. Idon'tknowwhy—bypurechanceoroneoftheseincrediblecoincidencesorpropheticinsights?—butIansweredjokingly,"Well,asyouknow,Johnny,Idon't knowmuchaboutengineeringorexperimentalphysics,infactIdon'tevenknowhowthetoiletflusherworks,exceptthatitisasortofautocatalyticeffect."AtthisI sawhimwinceandhisexpressionbecomequizzical.OnlylaterdidIdiscoverthatindeedthewordautocatalyticwasusedinconnectionwithschemesforthe constructionofanatomicbomb. Thenanothercoincidenceoccurred.Isaid,"RecentlyIhavebeenlookingatsomeworkonbranchingprocesses."TherewasapaperbyaSwedishmathematician aboutprocessesinwhichparticlesmultiplyquitelikebacteria,forexample.Itwasprewarwork,andaneleganttheoryofprobabilisticprocesses.That,too,could havehadsomethingtodowiththemathematicsofneutronmultiplication.Andagainhelookedatmealmostwithsuspicionorwonderandsmiledwanly. TheWisconsinastronomerJoelStebbins,whomIsawoccasionally,hadtoldmeaboutsomeworkgoingonwithuraniumandaboutthereleaseofenergyfromvery heavyelements.Iwonderedif,subconsciously,thispromptedmyremarks. Duringthismeetingatthestation,JohnnyandIalsodiscussedwhatseemedagenerallackofimaginationinthescientificcommunity'splanningofworkusefulforthe wareffort—especiallyincomputationsforhydrodynamicsandaerodynamics.Ipointedoutmydoubtsabouttheageofsomeofthemainparticipants(peopleover forty­fiveseemedtomeatthattimeold).Johnnyagreedtherewereobviousel­

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ementsofsenility.Asusual,wetriedtolightenoursadnesswithjocularcomments,observingthatsomeoneshouldestablisha"gerontological"society,whose memberswouldbescientistsinterestedinwarworkandafflictedwithprematureor"galloping"senility. SinceJohnnycouldnotorwouldnottellmewherehewasgoingexceptthatitwastotheSouthwest,IrememberedanoldJewishstoryabouttwoJewsonatrainin Russia.Oneaskstheother,"Whereareyougoing?"andthesecondreplies,"ToKiev."Whereuponthefirstsays,"Youliar,youtellmeyouaregoingtoKievsoI wouldthinkyouaregoingtoOdessa.ButIknowyouaregoingtoKiev,sowhydoyoulie?"AndItoldJohnny,"Iknowyoucan'ttellme,butyousayyouaregoing SouthwestinorderthatIshouldthinkthatyouaregoingNortheast.ButIknowyouaregoingSouthwest,sowhydoyoulie?"Helaughed.Wetalkedawhilelonger aboutthewarsituation,politics,andtheworld;thenhistwocompanionsreappearedandheleft. Isawhimoncemore,Ithink,inChicago,beforeIreceivedanofficialinvitationtojoinanunidentifiedprojectthatwasdoingimportantwork,thephysicshaving somethingtodowiththeinteriorofstars.TheletterinvitingmewassignedbythefamousphysicistHansBethe.Itcametogetherwithaletterfromthepersonnel departmentwithdetailsoftheappointment,salary,clearanceprocedures,indicationsonhowtogetthere,andsoforth.Iacceptedimmediatelywithexcitementand eagerness. Thepaywasslightlyabovemyuniversitysalary,butonatwelve­monthbasis—around$5,000,ifIremembercorrectly.TheprofessionalphysicistslikeBethewho weretherealreadyreceivedlittlemorethantheiruniversitysalaries.IlearnedlaterthatachemistfromHarvard,GeorgeKistiakowski,hadtheallegedlyastronomical salaryof$9,000or$10,000. Iinformedmyuniversityofthisopportunitytojoinanobviouslyimportantwarprojectandsecuredaleaveofabsencefortheduration.

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Astudentofmine,JoanHinton,hadleftforanunknowndestinationafewweeksbefore.JoanwastakingacourseIgaveinclassicalmechanics.Onedayshe appearedinmyofficeinNorthHalltoaskifIcouldgiveheranexaminationthreeorfourweeksbeforetheendofthetermsothatshecouldstartsomewarwork. SheproducedaletterfromProfessorIngraham,thechairman,authorizingmetodothat.Shewasagoodstudent,arathereccentricgirl,blonde,sturdy,good­looking. HerunclewasG.I.Taylor,theEnglishphysicist.Shewasalsoagreat­granddaughterofGeorgeBoole,thefamousnineteenth­centurylogician.Iwroteanumberof questionsonthebackofanenvelope;Joantooksomesheetsofpaper,satdownonthefloorwithhernotebook,wroteoutherexam,passed,anddisappearedfrom Madison. Soonafter,otherpeopleIknewwellbegantovanishoneaftertheother,withoutsayingwhere—cafeteriaacquaintances,youngphysicsprofessorsandgraduate studentslikeDavidFrisch,andhiswifeRose,whowasagraduatestudentinmycalculusclass,JosephMcKibben,DickTaschek,andothers. FinallyIlearnedthatweweregoingtoNewMexico,toaplacenotfarfromSantaFe.NeverhavingheardaboutNewMexico,Iwenttothelibraryandborrowed theFederalWriters'ProjectGuidetoNewMexico.Atthebackofthebook,ontheslipofpaperonwhichborrowerssignedtheirnames,IreadthenamesofJoan Hinton,DavidFrisch,JosephMcKibben,andalltheotherpeoplewhohadbeenmysteriouslydisappearingtohush­hushwarjobswithoutsayingwhere.Ihad uncoveredtheirdestinationinasimpleandunexpectedfashion.Itisnexttoimpossibletomaintainabsolutesecrecyandsecurityinwartime. Thisremindsmeofanotherstory.SinceIknewStebbinswell,aboutamonthafterarrivingatLosAlamos,Iwrotetohim.IdidnotsaywhereIwasbutmentioned thatinJanuaryorFebruaryIhadseenthestarCanopusonthehorizon.Lateritoccurredtomethatasanastronomerhecould

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easilyhavededucedmylatitudesincethisstaroftheSouthernskiesisnotvisibleabovethe38thparallel. Ishallpassoverourproblemsingettingtrainreservations.EvenwiththeprioritiesIhad,ourdeparturewasdelayedbyaboutamonth.OnthetrainIhadtooffera gratuitytotheconductortoobtainaberthforFrançoise,whowastwomonthspregnantatthetime.Thiswasthefirst—andIthinklast—timeinmylifethatI''bribed" anyone. Wearrivedataremote,lonely,unimpressivelittlewhistle­stop—Lamy,NewMexico.Tomyinfinitesurprise,theretomeetuswasJackCalkin,amathematicianI knewwell.IhadmethimseveralyearsbeforeattheUniversityofChicagoandhadseenhimanumberoftimessince.CalkinhadbeenJohnny'sassistantandhad gonewithhimtoLondontodiscussprobabilityproblemsinaerial­bombingpatternsandmethods.Justafewweeksbefore,hehadjoinedtheManhattanProject.He wasatall,pleasant­lookingman,withmoresavoir­fairethanmostmathematicians.HavingheardthatIwascoming,heborrowedacarfromtheArmymotorpooland drovetomeetusatthetrain. Thesunshonebrilliantly,theairwascrispandheady,anditwaswarmeventhoughtherewasalotofsnowontheground—alovelycontrasttotherigorsofwinterin Madison.CalkindroveusintoSantaFe,andwestoppedforlunchattheHotelLaFonda,wherewesatatthelowSpanish­styletablesinthebar.Afteraninteresting New­Mexico­stylemeal,wewalkedtoasmalldoorwayinaone­storybuildingonalittlestreetthatborderedthecentralPlaza.Inamodestsuiteofrooms,asmiling middle­agedladyinvitedmetofilloutafewforms,turnedacrankonaprimitivedeskmachine,andproducedthesheetsofpaperthatwereourpassestotheLos AlamosProject.ThisinconspicuouslittleofficewastheentrancetothegiganticLosAlamoscomplex.Thescene,verymuchlikeaBritishcloak­and­daggermystery, broughtmemoriesofmyboyhoodfascinationwithsuchtales. TheprojectsitewasaboutfortymilesnorthwestofSantaFe.Theridewashair­raising,Jackhavingelectedtoshow

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usthecountrysidebytakingusonashortcut—amuddytrackthroughsparseMexicanandIndianvillages—untilwecametotheRioGrande,whichwecrossedona narrowwoodenbridge. Thesettingwasromantic.Weweregoingupandupintoastrange,mysteriouslandscapeofmesas,cliffs,piñontreesandbrush.Thisbecameaforestofpinesaswe gainedelevation.Atamilitarygateinabarbed­wirefence,weshowedourpassesanddroveontoahelter­skeltercollectionofone­andtwo­storywoodenstructures builtalongmuddy,unpaved,narrowstreetsandpaths. Wewereassignedasmallcottagebyapond(withthepromiseoflargerquartersassoonastheywerebuilt).IthenfollowedJacktomyfirstvisittothetechnicalarea. Weenteredanoffice,whereIwassurprisedtofindJohnnydeepinconversationwithamanofmiddlestature,bushyeyebrows,anintenseexpression.Helimped slightlyashepacedbackandforthinfrontofablackboard.ThiswasEdwardTeller,towhomJohnnyintroducedme. TheyweretalkingaboutthingswhichIonlyvaguelyunderstood.Thereweretremendouslylongformulaeontheblackboard,whichscaredme.Seeingallthese complicationsofanalysis,Iwasdumbfounded,fearingIwouldneverbeabletocontributeanything.However,whendayafterdaythesameequationsremainedand werenotchangedeveryfewhoursasIhadexpected,Iregainedmyconfidenceandsomehopeofbeingabletoaddsomethingtothetheoreticalwork. Iunderstoodsnatchesoftheirconversation,andanhourlater,Johnnytookmeasideandexplainedtomeformallyandclearlythenatureoftheprojectanditsstatusat themoment.TheworkinLosAlamoshadstartedinearnestonlyabouttwoorthreemonthsbefore.VonNeumannseemedverycertainofitsimportanceandradiated confidenceabouttheultimatesuccessoftheenterprisewhoseobjectivewastheconstructionofanatomicbomb.Hetoldmeofallthepossibilitieswhichhadbeen considered,ofthe

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problemsrelatingtotheassemblingoffissionablematerials,aboutplutonium(whichdidnotyetphysicallyexisteveninthemostmicroscopicquantitiesatLosAlamos). Irememberverywell,whenacoupleofmonthslaterIsawRobertOppenheimerrunningexcitedlydownacorridorholdingasmallvialinhishand,withVictor Weisskopftrailingafterhim.Hewasshowingsomemysteriousdropsofsomethingatthebottomofthevial.Doorsopened,peopleweresummoned,whispered conversationsensued,therewasgreatexcitement.Thefirstquantityofplutoniumhadjustarrivedatthelab. Needlesstosay,IsoonranacrossmostoftheWisconsiniteswhohadsomysteriouslydisappearedfromMadisonbeforeus.ImetHansBetheonthefirstday.I knewmoreabouthimthanaboutTeller.Igraduallymettheentiregroupoftheoreticalandexperimentalphysicists.IhadknownmanymathematiciansinEuropeandin thiscountry,butnotasmanyphysicists. Ihadsomeknowledgeoftheoreticalphysics,despitethejokeIhadtoldJohnnyaboutmynotunderstandingeventheautocatalyticactionofatoilet.Astronomy,of course,hadbeenmyfirstinterest,andthenphysicsandmathematics.IhadevengivenacourseinclassicalmechanicsatHarvard,butitisonethingtoknowabout physicsabstractly,andquiteanothertohaveapracticalencounterwithproblemsdirectlyconnectedwithexperimentaldata,suchastheverynoveltechnologywhich wastocomefromLosAlamos. Ifoundoutthatthemainabilitytohavewasavisual,andalsoanalmosttactile,waytoimaginethephysicalsituations,ratherthanamerelylogicalpictureofthe problems. Thefeelingforproblemsinphysicsisquitedifferentfrompurelytheoreticalmathematicalthinking.Itishardtodescribethekindofimaginationthatenablesoneto guessatorgaugethebehaviorofphysicalphenomena.Veryfewmathematiciansseemtopossessittoanygreatdegree.Johnny,forexample,didnothavetoany extenttheintuitivecommonsenseand"gut"feelingorpenchantforguessing

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whathappensingivenphysicalsituations.Hismemorywasmainlyauditory,ratherthanvisual. Anotherthingthatseemsnecessaryistheknowledgeofadozenorsophysicalconstants,notmerelyoftheirnumericalvalue,butarealfeelingfortheirrelativeorders ofmagnitudeandinterrelations,and,sotospeak,aninstinctiveabilityto"estimate." Iknew,ofcourse,thevaluesofconstantslikethevelocityoflightandmaybethreeorfourotherfundamentalconstants—thePlanckconstanth,agasconstantR,etc. VerysoonIdiscoveredthatifonegetsafeelingfornomorethanadozenotherradiationandnuclearconstants,onecanimaginethesubatomicworldalmosttangibly, andmanipulatethepicturedimensionallyandqualitatively,beforecalculatingmorepreciserelationships. MostofthephysicsatLosAlamoscouldbereducedtothestudyofassembliesofparticlesinteractingwitheachother,hittingeachother,scattering,sometimesgiving risetonewparticles.Strangelyenough,theactualworkingproblemsdidnotinvolvemuchofthemathematicalapparatusofquantumtheoryalthoughitlayatthebase ofthephenomena,butratherdynamicsofamoreclassicalkind—kinematics,statisticalmechanics,large­scalemotionproblems,hydrodynamics,behaviorofradiation, andthelike.Infact,comparedtoquantumtheorytheprojectworkwaslikeappliedmathematicsascomparedwithabstractmathematics.Ifoneisgoodatsolving differentialequationsorusingasymptoticseries,oneneednotnecessarilyknowthefoundationsoffunctionspacelanguage.Itisneededforamorefundamental understanding,ofcourse.Inthesameway,quantumtheoryisnecessaryinmanyinstancestoexplainthedataandtoexplainthevaluesofcrosssections.Butitwasnot crucial,onceoneunderstoodtheideasandthenthefactsofeventsinvolvingneutronsreactingwithothernuclei. Teller,inwhosegroupIwassupposedtowork,talkedtomeonthatfirstdayaboutaprobleminmathematicalphys­

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icsthatwaspartofthenecessarytheoreticalworkinpreparationfordevelopingtheideaofa"super"bomb,astheproposedthermonuclearhydrogenbombwasthen called.Theideaofthermonuclearreactionsthatwouldreleaseenormousamountsofenergywas,ofcourse,older.Theirroleinthereactionsintheinteriorofstarswas discussedintheoreticalpapersinthe1930sbythephysicistsGeoffreyS.AtkinsonandFritzHoutermans.Theideaofusingauraniumfissionexplosiontotriggera thermonuclearreactioncanbecreditedtoTeller,Bethe,Konopinski,Ibelieve,andperhapssomeothers. Teller'sproblemconcernedtheinteractionofanelectrongaswithradiation,andithadmoretodowiththethermonuclearpossibilitiesthanwiththeassemblyofthe fissionbomb,whichwasthemainproblemandworkofLosAlamos.Heguessedaformulaforenergytransfersconnectedwiththeso­calledComptoneffectabout therateofenergytransfer.Thisformula,basedondimensionalgroundsandhisintuitionalone,wasquitesimple;heaskedmetotrytoderiveitmorerigorously.Asit waspresented,therewasnonumericalfactorinfront.Thisseemedcurioustome.Iaskedhimexplicitlyaboutthisadayortwolater,andhesaid"Oh,thefactor shouldbe1." ThiswasthefirsttechnicalproblemintheoreticalphysicsIhadevertackledinmylife,andIapproacheditfromaveryelementarypointofview.Ireadpaperson statisticalmechanics,onpropertiesoftheradiationfield,andstartedworkingwithrathernaiveandcommon­sensekinematicpictures.Itriedsomearithmeticand obtainedaformulamuchlikeTeller's,butwithanumericalfactorofaboutfourinfrontastherateoftransfer.Itwasamessylittlejob.Edwardwasnotsatisfiedwith myratherelementaryderivations. ShortlyafterIhaddiscussedthisworkwithTeller,ayoung,moreprofessionalmathematicalphysicist,HenryHurwitz,Jr.,joinedTeller'sgroupandwithhismuch bettermathematicaltechniquesandexperienceinthespecialfunctionsthatwereusedinthistypeofproblem,heob­

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tainedaformula,muchmorescholarlythanmine,involvingBesselfunctions.Indeed,theexactnumericalfactorwasnotverydifferentfromfour.IfIremember correctly,itwasarootofacertainBesselfunction. Theideawastohavesomethermonuclearmaterial—deuterium—nexttothefissionbomb,andtoletitigniteaftertheuraniumbombhadexploded.Howtodoitin detailwasthebigproblem,anditwasbynomeanseasytoseehowsuchanarrangementwouldigniteandnotjustsputterandfizzleout.Therewasalso,theoretically atleast,thehazardofgettingmoreofanexplosionthanintendedandofhavingthewholeatmosphereoftheearthignite!Thewell­knownphysicistGregoryBreitwas involvedincalculatingthechancesoftheignitionoftheatmosphere.These,ofcourse,hadtobezerobeforeonecouldeventhinkoftamperingwiththermonuclear reactionsonearth. IthinkitwasBethe,withEmilKonopinski,awell­knowntheoreticalphysicist,whosuggestedtritiuminsteadofdeuteriumasamaterialeasiertoignite,giventhe temperatureofthefissionbomb.Suchanengineeringsuggestionfromtheoreticalworkcamefromhissuperbknowledgeoftheoreticalnuclearphysics. Bethewastheheadofthetheoreticaldivision,asitwascalled.ActuallyitwashisandRobertF.Bacher'spapersinReviewsofModernPhysicswhichwereusedas the"bible"oftheLosAlamosscientists,fortheycontainedthebulkofthetheoreticalideasandexperimentalfactsknownatthetime.Bethe,nowaNobelPrize­ winnerforhisearlierdiscoveryofthemechanismofenergygenerationinthesunandotherstars(theso­calledcarboncycle),is,amongotherthings,avirtuosointhe techniquesofmathematicalphysics.AsFeynmanonceputit,atLosAlamoshewas,withhisrigorousanddefinitivework,likeabattleshipmovingsteadilyforward, surroundedbyaflotillaofsmallervessels,theyoungertheoreticalworkersofthelaboratory.HeisoneofthefewpersonsforwhomImerelyhadrespectinthe

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beginningbutovertheyearshavecontinuouslydevelopedlikingandadmiration. WhenIfirstmetTeller,heappearedyouthful,alwaysintense,visiblyambitious,andharboringasmolderingpassionforachievementinphysics.Hewasawarmperson andclearlydesiredfriendshipwithotherphysicists.Possessingaverycriticalmind,healsoshowedquickness,sense,andgreatdeterminationandpersistence. However,Ithinkhealsoshowedlessfeelingfortruesimplicityinthemorefundamentallevelsoftheoreticalphysics.Toexaggerateabit,Iwouldsayhistalentswere moreinthedirectionofengineering,construction,andthesurveyingofexistingmethods.Butundoubtedlyhealsohadgreatingenuity. Tellerwaswellknownforhisworkonmolecules,buthemayhaveconsideredthisasasortofsecondaryfield.IthinkitwastheeasewithwhichGamowhadnew ideaswithoutanytechnicalarsenalathisdisposalthatpushedTellerintotryingtoemulatehimandtoattemptmorefundamentalwork. AfterhegotintopersonaldifficultieswithTellerontheorganizationalfeaturesofthehydrogenwork,GamowlatertoldmethatbeforethewarTellerwas,inhisview, adifferentperson—helpful,willing,andabletoworkonotherpeople'sideaswithoutinsistingoneverythinghavingtobehisown.AccordingtoGamow,something changedinhimafterhejoinedtheLosAlamosProject. Ofcourse,manyphysicistswhowerealmostcongenitallyivory­towertypesgottheirheadsturnedwiththesuddenrealizationofnotonlythepracticalbutworldwide historicalimportanceoftheirwork—nottomentionthemoretrivialbutobviousmatteroftheenormoussumsofmoneyandphysicalfacilitiesthatsurpassedanythingin theirpreviousexperience.Perhapsthisplayedaroleinthepersonalitychangeofsomeprincipals;withOppenheimer,thedirector,itmayhavehadabearingonhis subsequentactivities,career,ideas,androleasauniversalsage.LikeTeller,Op­

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penheimermayhavehadafeelingofinadequacyascomparedwiththecreatorsofgreatnewphysicaltheories.Hewasequaltoorevenmorebrilliantandquickthan Teller,butperhapslackedtheultimatecreativesparkoforiginality.Withhisfantasticintelligence,hemusthaverealizedthishimself.Inspeedofunderstandingandin criticalability,heprobablysurpassedBetheorFermi. TellerwantedtohavehisownstamponmuchoftheessentialworkofLosAlamos,atfirstviahisownapproachtothefissionbomb.Hewaspushingformilder explosions,dilutionofmaterial,etc.InspiteofcalculationsbyKonopinskiandothersthatgaveapooroutlookforsomeoftheseplans,hewastryingbyeverymeans tohavehisownadopted.CollaborationwithBethe,whowasheadofthetheoreticaldivision,becameincreasinglydifficult. AsdisagreementsbetweenTellerandBethebecamemorefrequentandacute,Tellerthreatenedtoleave.Oppenheimer,whodidnotwanttolosesuchabrilliant scientist,agreedtolethimandhisgroupworkinamorefuture­orientedfield,independentoftheproject'smainline.ThisishowTellerbegantoconcentrateand organizethetheoreticalworkforthe"Super."Konopinski,Weisskopf,Serber,RichardFeynman,WilliamRarita,andmanyothersallhadspecialcontributionsto make,butitwasreallyTellerwhokeptthethingtogetherandmovingforwardduringthewar. AfterFermi'sarrival,Teller'sgroupbecameapartofFermi'sdivision.FermitookgreatinterestinthetheoreticalworkonthethermonuclearreactionsandH­bomb possibilities;attheendofthewarhegaveaseriesoflecturesmagnificentlysummarizingtheworkdoneuntilthen—thanksmostlytotheinvestigationsofTellerandhis group. Buteventhen,beforethesuccessoftheatomicbombitself,someofTeller'sactionsgavecluestowhatledtomuchoftheunpleasantnessandwasteoftimeintheso­ calledH­Bombcontroversy. Teller'sgroupwascomposedofanumberofveryinterestingyoungphysicists,youngereventhanTeller,Kono­

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pinski,ormyself.ItincludedNickMetropolis,aGreek­Americanwithawonderfulpersonality;HaroldandMaryArgo,ahusband­and­wifeteam,eagerandtalented; JaneRoeberg,ayoungwomanwhogavetheimpressionofbeingcompetent;andafewotherswhosenamesIdonotnowremember. Therewas,ofcourse,muchcontactwithothergroupsofphysicistswhowereontheborderlineofproblemsconcerningthepossibilitiesofa"super";discussionswith themwerefrequent,pleasant,andconcernedmanydifferentbranchesofphysics.Onecouldhearabouttheprosandconsoftheideaofimplosion,whichwasnew andvigorouslydebatedinmanyoftheoffices.Thesediscussionswerecompletelyopen.Nothingwasconcealedfromanybodywhowasascientist. Themoreformalwayoflettingpeopleknowwhatwasgoingonwastheweeklycolloquia,whichwereheldinabighangarthatalsoservedasthemovietheater. Thesetalkscoveredprogressoftheworkofthewholelaboratoryandthespecificproblemswhichtheprojectencountered.TheywererunbyOppenheimerhimself. Asformyself,afterthisfirstworkonEdward'sproblem,Ispreadoutmyintereststootherrelatedquestions,onebeingtheproblemofstatisticsofneutron multiplication.Thiswasmoretangibleformefromthepurelymathematicalside.IdiscussedsuchproblemsofbranchingandmultiplyingpatternswithDavidHawkins. Wewroteareportonmultiplicativebranchingprocesses,whichhadsomepracticalapplicationandrelevancetotheproblemoftheinitialdetonationofthebombbya fewneutrons.ThisproblemwasalsostudiedbyStanFrankelandbyFeynman,inamoretechnicalandclassicalway.Ourpapercouldbeconsideredthebeginningof whatwouldcometobeknowninmathematicsasbranchingprocessestheory,asub­fieldofprobabilitytheory. IalsotalkedalotwithvonNeumannandCalkinaboutproblemsofhydrodynamics,especiallythoseconcerningthe

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processofimplosion.SomewhattomysurpriseIfoundmypurelyabstractintellectualhabitsasamathematicianimmediatelyusefulintheworkwiththesemore practical,special,andtangibleproblems.Ihaveneverfeltthe"gap"betweenthemodeofthinkinginpuremathematicsandthethinkinginphysics,onwhichmany mathematiciansplacesomuchstress.Anythingamenabletomentalanalysiswascongenialforme.Idonotmeanthedistinctionbetweenrigorousthinkingandmore vague"imaginings";eveninmathematicsitself,allisnotaquestionofrigor,butrather,atthestart,ofreasonedintuitionandimagination,and,also,repeatedguessing. Afterall,mostthinkingisasynthesisorjuxtapositionofadvancesalongalineofsyllogisms—perhapsinacontinuousandpersistent"forward''movement,with searching,sotospeak"sideways,"indirectionswhicharenotnecessarilypresentfromtheverybeginningandwhichIdescribeas"sendingoutexploratorypatrols" andtryingalternativeroutes.Itisallamulticoloredthing,notveryeasytodescribeinawaythatareadercanappreciate.ButIhopethiskindofpersonalanalysisof thinkinginscienceisoneofthepossibleinterestsofthisbook. AdiscussionwithvonNeumannwhichIrememberfromearly1944tookseveralhours,andconcernedwaystocalculatethecourseofanimplosionmorerealistically thanthefirstattemptsoutlinedbyhimandhiscollaborators.Thehydrodynamicalproblemwassimplystated,butverydifficulttocalculate—notonlyindetail,buteven inorderofmagnitude. Inparticular,thequestionsconcernedvaluesofcertainnumbersrelatingtocompressionversuspressure,andsuch.Thesehadtobeknown,letussaywithintenper centorbetter,butthesimplificationsmadeintheoutlineofthecalculationswereofsuchanaturethattheycouldnotguaranteeaccuracywithinafactoroftwoor three.AlltheingeniousshortcutsandtheoreticalsimplificationswhichvonNeumannandothermathematicalphysicistssuggested,andwhichhetriedtoexecutewith thehelpofCalkin,seemed

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inadequatetome.InthisdiscussionIstressedpurepragmatismandthenecessityforattemptingtogetaheuristicsurveyofthegeneralproblembysimplemindedbrute force—thatis,morerealistic,massivenumericalwork.Atthattime,in1944,withtheavailablecomputingfacilities,theaccuracyofthenecessarynumericalwork couldnotbesatisfactory.Thiswasoneofthefirstreasonsforpressingforthedevelopmentofelectroniccomputers. OneofthecharmsandgreatattractionsoflifeinLosAlamosinthosedayswasthelunchesattheLodge,inthemidstoffriends.Iwasverysurprisedtofindthereand graduallytomeetsomanyfamouspersonsIhadheardabout. LosAlamoswasaveryyoungplace.Atthirty­four,Iwasalreadyoneoftheolderpeople.Whatimpressedmemostwastheverygreatcompetenceoftheyounger peopleandthevarietyoftheirfieldsofspecialization.Itwasalmostlikehavinganencyclopediatolookat,somethingthatIsomuchliketodo.Ihadthesamefeeling whentalkingtotheyoungscientistsaroundthelaboratory.Itisnottherightexpressionperhapsbut,roughlyspeaking,theyweremoreaccomplishedindepththanin breadth.Theoldermen,manyofwhomwereEuropean­born,hadamoregeneralknowledge.Yetsciencehadbecomesoramified,specializationhadproceededso far,thatitwasquitedifficulttoretainknowledgeofallthedetailsandtheoverallviewatthesametime. Theyoungerscientistsshowedalotofcommonsenseintheirownfields,butingeneralagreathesitationtoengageinspeculationoutsidetheirareas.Perhapsthis stemmedfromafearofnotbeing"absolutelyright."Manydisplayedacertainanti­philosophicalspirit—notanti­intellectual,butanti­philosophical.Thiswasperhaps becauseofthepragmaticnatureofAmericanattitudes. Iwasalsostruckbythewell­knownAmericantalentforcooperation,theteamspirit,andhowitcontrastedwithwhatIhadknownincontinentalEurope.I rememberedhowJulesVernehadanticipatedthiswhenhewroteabout

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thecollectiveeffortneededfortheorganizationofhis"VoyagetotheMoon."Peopleherewerewillingtoassumeminorrolesforthesakeofcontributingtoacommon enterprise.Thisspiritofteamworkmusthavebeencharacteristicoflifeinthenineteenthcenturyandwaswhatmadethegreatindustrialempirespossible.Oneofits humoroussideeffectsinLosAlamoswasafascinationwithorganizationalcharts.Atmeetings,theoreticaltalkswereinterestingenoughtotheaudience,butwhenever anorganizationalchartwasdisplayed,Icouldfeelthewholeaudiencecometolifewithpleasureatseeingsomethingconcreteanddefinite("Whoisresponsibleto whom,"etc.).OrganizationwasandperhapsstillisagreatAmericantalent,althoughthisiswrittenatatimewhentheso­calledenergycrisisappearstometobemore acrisisofmomentumthanofenergy(acrisisofenterprise,solidarity,commonspirit,determination,andcooperationforthecommongood). Itisdifficulttodescribeforthegeneralreadertheintellectualflavor,thefeeling,ofascientific"atmosphere."ThereisnospecificEnglishwordforthisimpression.Odor andsmellhaveunpleasantconnotations;perfumeisartificial;auraissuggestiveofmystery,ofthesupernatural.Theyoungerscientistsdidnothavemuchofanaura, theywerebrightyoungmen,notgeniuses.PerhapsonlyFeynmanamongtheyoungoneshadacertainaura. SixorsevenyearsyoungerthanI,hewasbrilliant,witty,eccentric,original.IrememberonedayBethe'slaughtershookthecorridorwalls,makingmerushoutofmy officetoseewhatwassofunny.ThreedoorsdowninBethe'soffice,Feynmanwasstanding—talkingandgesticulating.Hewastellingthestoryofhowhehadfailed hisdraftphysicalexamination,re­enactinghisnowfamousgesture:whenadoctoraskedhimtoshowhishands,hechosetostretchtheminfrontofhimonewithpalm up,theotherpalmdown.Thedoctorsaid,"Theotherside!"andhereversedbothhands.Thisandotherincidentsofhisphysicalexaminationhadcausedanexplosion oflaughterontheentirefloor.Itwason

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thefirstorseconddayinLosAlamosthatImetFeynman,andremarkedtohimaboutmysurprisethatE=mc —whichIofcoursebelievedintheoreticallybut somehowdidnotreally"feel"—was,infact,thebasisofthewholethingandwouldbringaboutabomb.WhatthewholeProjectwasworkingondependedonthose fewlittlesignsonpaper.Einsteinhimself,whenhewasfirsttoldbeforethewaraboutradioactivephenomenashowingtheequivalenceofmassandenergy,allegedly replied,"Istdaswirklichso?istdaswirklichso?"(Isthatreallyso?) JokinglyItoldFeynman,"Onedaypeoplewilldiscoverthatacubiccentimeterofvacuumisreallyworthtenthousanddollars—itisequivalenttosomuchenergy."He immediatelyagreedandadded,"Yes,butofcourseitwillhavetobepurevacuum!"Indeed,peoplenowknowaboutthepolarizationofvacuum.Theforcebetween twoelectronsortwoprotonsisnote2/r2,butaninfiniteseriesofwhichthisisthefirstterm.Itworksonitself,liketwoalmost­parallelmirrors,whichshowareflection ofareflectionofareflection,adinfinitum. WritingthisremindsmeofafeelingIoncehadwhenIvisitedthecyclotroninChicagowithFermi.Hetookmearoundandmademewalkthroughanincrediblyheavy door,whichhesaid"wouldflattenyouintoapieceofpaperwereitclosedonyou."Wewalkedbetweenthepolesofthemagnet,andIreachedintomypocketfor thepenknifewithwhichIsometimesplay.SuddenlyitwasjerkedoutofmyhandwhenItouchedit.Thepowerofthevacuum!Thismademephysicallyconsciousof therealityof"empty"space. Feynmanwasalsointerestedinmanypurelymathematicalrecreationsnotrelatedtophysics.Irememberhowheoncegaveanamusingtalkabouttriangularnumbers andmanagedtoentertaineverybodywithhishumor.Atthesametime,hewasdoingmathematics,andshowingthefoolishnessofexcessiveclevernessandthe irrationalityofsuchstrangeinterests.

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Onedayherecitedtomethefollowing: "IwonderwhyIwonder, IwonderwhyIwonderwhyIwonder. IwonderwhyIwonderwhyIwonderwhy,"andsoon. Italldependsonwhereyouputtheintonation,conveyingadifferentmeaningineverycase.Hediditmarvelouslyinfiveorsixstyles,eachwithdifferentstresses,asit were,mosthumorously. Physically,LosAlamosconsistedofacollectionoftwo­andfour­apartmentbuildings,temporaryArmystructureswhichturnedouttobesturdyenoughtosurvivefor manyyearsaftertheendofthewar.Tohiseverlastingcredit,Oppenheimerinsistedthattheybelaidoutalongthecontoursoftheland,retainingasmanytreesas possible,insteadofinthemonotonousrectangularpatternofarmycampsandcompanytowns.Stilltheywereratherprimitive,equippedwithcoalfurnacesandcoal stovesinthekitchens.Peoplegripedabouttheinadequaciesofthehousingsituation,andwiveshadallsortsofcomplaints.ButIfoundLosAlamosonthewholequite comfortable.TheclimateofNewMexico—LosAlamosinparticular,atanelevationofseventy­twohundredfeet—wasoneofthebestIhaveeverlivedin. Placzek,aphysicistwhojoinedtheprojectafterthewar,feltthateastoftheRockyMountainstheUnitedStateswasonthewholeclimaticallyuninhabitable, "unbewohnbar."ThisistrueespeciallyforEuropeanswhoarenotaccustomedtohotandmuggysummersortopenetratingwintercold.InCambridge,Iusedtotell myfriendsthattheUnitedStateswaslikethelittlechildinafairytale,atwhosebirthallthegoodfairiescamebearinggifts,andonlyonefailedtocome.Itwastheone bringingtheclimate. SoonaftermyarrivalinLosAlamosImetDavidHawkins,ayoungphilosopherfromBerkeley,oneofthepeopleOppenheimerhadbroughtwithhimtostaffthe administrationoftheLaboratory.Wehititoffintellectuallyrightaway. Hawkinsisatallish,blue­eyed,blonddescendantof

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earlyNewMexicosettlers.Hisfather,JudgeHawkins,wasafamousfigureattheturnofthecentury.HewasalawyerandanofficialoftheTerritory,importantinthe SantaFeRailroadoperations.DavidwasbroughtupinthesmallcommunityofLaLuz,inthesouthernpartoftheState.Imentionthisbecauselater,whenthebomb wasexplodedintheJornadadelMuertedesertnearAlamagordo,Davidworriedthatblindingflashesortheheatandshockphenomenamightbedangerousfor peoplelivinginLaLuz,somethirtyorfortymilesaway,wherehissisterhadherhome. Hawkinsisamanofwideinterests,withgreatbreadthofknowledge,verygoodeducation,andaverylogicalmind.Heregardsscientificproblemsnotasanarrow specialist,butfromageneralepistemologicalandphilosophicalpointofview.Totopitoff,heisthemosttalentedamateurmathematicianIknow.Hetoldmethatat StanfordhetooksomecoursesfromOuspenski,theRussianémigréspecialistinprobabilityandnumbertheory,buthehasnothadanyextensivetrainingin mathematics.Hehasaverygreatnaturalfeelingforitandatalentformanipulation.Heisthemostimpressiveofthenon­professionalmathematiciansorphysicistsI havemetanywhereintheworld. Wediscussedproblemsofneutronchainreactionsandtheprobabilityproblemsofbranchingprocesses,ormultiplicativeprocesses,aswecalledthemin1944. Iwasinterestedinthepurelystylizedproblemofabranchingtreeofprogenyfromoneneutronwhichmaymultiply,intozero(thatis,thedeathofaneutronby absorption),orone(thatjustcontinuesitself),ortwoorthreeorfour(thatis,causestheemergenceofnewneutrons),eachpossibilitywithagivenprobability.The problemistofollowthefuturecourseandthechainofpossibilitiesthroughmanygenerations. VeryearlyHawkinsandIdetectedafundamentaltricktohelpstudysuchbranchingchainsmathematically.Theso­calledcharacteristicfunction,adeviceinventedby Laplace

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andusefulfornormal"addition"ofrandomvariables,turnedouttobejustthethingtostudy"multiplicative"processes.Laterwefoundthatobservationstothiseffect hadbeenmadebeforeusbythestatisticianLotka,buttherealtheoryofsuchprocesses,basedontheoperationofiterationofafunctionorofoperatorsalliedtothe function(amoregeneralprocess),wasbegunbyusinLosAlamos,startingwithashortreport.Thisworkwasstronglygeneralizedandbroadenedin1947,afterthe war,byEverettandmyselfafterhejoinedmeinLosAlamos.Sometimelater,EugeneWignerbroughtupaquestionofpriorities.Hewaseagertonotethatwedid thisworkquiteabitbeforethecelebratedmathematicianAndreiN.KolmogoroffandotherRussiansandsomeCzechshadlaidclaimtohavingobtainedsimilar results. IlikedHawkins'sgeneralcuriosity,hisalmostuniqueknowledgeofthefundamentalsofseveralscientifictheories—notonlyintheconceptualelementsofphysicsbut inbiologyandeveneconomics.Ilikedhisinterestandgenuinelyoriginalworkinwhatwastobecomeknownas"informationtheory"afteritbecameformalizedby WienerandespeciallybyClaudeShannon.DavidappliedtoeconomicproblemsthemathematicalideasofvonNeumannandMorgensterningametheory. Hawkinshassincewrittenseveralinterestingpapersandanexcellentbookonthephilosophyofscience,orratheronthephilosophyofrationalthinking,calledThe LanguageofNature. Hawkins'spositioninLosAlamosatfirstwasasaliaisonbetweenOppenheimer'sofficeandthemilitary.Someyearslaterhewrotetwovolumes,sincedeclassified, abouttheorganizationandthescientifichistoryfromtheearlydaysofLosAlamosuntiltheendofthewar.Ididnotknowitatthetime(anditwasnotobviousfrom conversationswithhim)thatinthenineteenthirtieshehadbeeninvolvedontheWestCoastwithcommunistsympathizergroups.Thatcausedhimgreattroublebefore andduringtheMcCarthy

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era,includinghearingsinWashington.Hecameoutofitcompletelyvindicated. Hiswife,Frances,anextremelyinterestingperson,becamefriendlywithFrançoise,andwesaweachotheragreatdeal.AtthetimeofmyillnessinCaliforniain1946, theHawkinseswereimmenselyhelpfultousincaringforourdaughterClaire,whowasthenaneighteen­month­oldinfant. HawkinsleftLosAlamosaftertheendofthewartotakethepostofprofessorofphilosophyattheUniversityofColoradoinBoulder,whereheistoday. TheLosAlamoscommunitywascompletelydifferentfromanywhereIhadeverlivedandworked.EvenLwów,whichhadadenseconcentrationofpeopleand wherethemathematiciansanduniversitypeoplewereindailycontactandspentmuchtimetogetherinrestaurantsandcoffeehouses,didnothavethedegreeof togethernessofLosAlamos.Itwasevenmorepronouncedtherebecauseoftheisolationandthesmallnessofthetown,andtheproximityofallthebuildings.People visitedeachotherconstantlyatallhoursafterwork.Whatwasnoveltomewasthatthesewerenotmathematicians(exceptvonNeumannandtwoorthreeyounger persons),butphysicists,chemists,andengineers—psychologicallyquitedifferentfrommymoreinward­orientedmathematicalcolleagues.Thevarietyandrichnessof thephysicistswasinterestinganddelightfultoobserve.Onthewhole,theoreticiansandexperimentalistsdifferedintemperament. IthasbeensaidthatatlunchinFullerLodgeonecouldseeasmanyaseightortenNobelPrize­winnerseatingatthesametime(Rabi,Lawrence,Fermi,Bloch,Bohr, Chadwick,andothers).Theirinterestswerewidebecausephysicshasmoredefiniteandobviouscentralproblemsthanmathematics,whichsplitsintomanyalmost independentdomainsofthought.Theyconsiderednotonlythemainproblem—theconstructionofanatomicbombandrelatedphysicalquestionsaboutphenomena thatwouldattendtheex­

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plosion—thestrictlyprojectwork—butalsogeneralquestionsaboutthenatureofphysics,thefutureofphysics,theimpactofnuclearexperimentsonthetechnology ofthefuture,andcontrastinglyitsinfluenceonthefuturedevelopmentoftheory.Beyondthis,Irememberverymanyafter­dinnerdiscussionsaboutthephilosophyof science,andofcourseontheworldsituation,fromdailyprogressonthewarfrontstoprospectsofvictoryinthemonthstocome. Theintellectualqualityofsomanyinterestingpersonsandtheirbeingconstantlytogetherwasunique.Intheentirehistoryofsciencetherehadneverbeenanything evenremotelyapproachingsuchaconcentration.TheradarprojectinCambridge,Massachusetts,proceedingatthesametime,hadsomeofthesecharacteristics,but withoutthesameintensity.Itwasmoretechnologicalperhaps,anddidnottouchasmanyfundamentalquestionsofphysics. Whoweresomeoftheluminariesofthisfantasticassembly?VonNeumann,Fermi,Bethe,Bohr,Feynman,Teller,Oppenheimer,O.R.Frisch,Weisskopf,Segré, andmanymore.Ihavealreadytriedtosketchthepersonalitiesofsomeofthemandcandescribeafewmore. IfirstmetFermiwhenhearrivedatLosAlamos,afewmonthsafterus,aftertheChicagopilehadbeensuccessfullycompleted.IremembersittingatlunchinFuller Lodgebeforehisarrivalwithsixorsevenpeople,includingvonNeumannandTeller.Tellersaid,"ItisquitecertainnowthatEnricowillarrivenextweek."Ihad learnedearlierthatFermiwasreferredtoas"thepope"becauseoftheinfallibilityofhispronouncements.SoimmediatelyIintoned:"Annunciovobisgaudium maximum,papamhabemus,"whichistheclassicalwaycardinalsannouncetheelectionofapopeonthebalconyoverlookingSt.Peter'sSquare,afterthewhitesmoke comesoutofthechimneyintheVatican.Johnny,whounderstood,explainedthisreference,andtheallusionwasapplaudedbytheentiretable. Fermiwasshort,sturdilybuilt,stronginarmsandlegs,andratherfastmoving.Hiseyes,dartingattimes,wouldbe

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fixedreflectivelywhenhewasconsideringsomequestion.Hisfingersoftennervouslyplayedwithapencilorasliderule.Heusuallyappearedingoodhumor,witha smilealmostperpetuallyplayingaroundhislips. Hewouldlookataquestionerinaninquiringway.Hisconversationincludedmanyquestionsratherthanexpressionsofopinion.Hisquestionswereformulatedinsuch away,however,thatitwasclearwhichwayFermi'sbeliefsorguesseswent.Hewouldtrytoelucidateotherpersons'thoughtsbyaskingquestionsinaSocratic manner,yetmoreconcretelythaninPlato'ssuccessionofproblems. Sublimatedcommonsensecharacterizedhisthoughts.Hehadwillpowerandcontrol;andnotobstinacybutpersistenceinfollowingaline,allthewhilelookingvery carefullyatpossibleramifications.Hewouldnotneglecttheopportunitiesthatpresentedthemselves,oftenbychance,fromrandomobservationsinscientificwork. Oncewhenwediscussedanotherphysicist,hecharacterizedhimastoosystematicallyobstinate.Yethealsotoldmethathelikedtoworkverysystematicallyinan orderlyfashioninordertokeepeverythingundercontrol.Atthesametimehehaddecidedinhisyouthtospendatleastonehouradaythinkinginaspeculativeway.I likedthisparadoxofasystematicwayofthinkingunsystematically.Fermihadawholearsenalofmentalpictures,illustrations,asitwere,ofimportantlawsoreffects, andhehadagreatmathematicaltechnique,whichheusedonlywhennecessary.Actuallyitwasmorethanmeretechnique;itwasamethodfordissectingaproblem andattackingeachpartinturn.Withourlimitedknowledgeofintrospectionthiscannotbeexplainedatthepresenttime.Itisstillan"art"ratherthana"science."I wouldsaythatFermiwasoverwhelminglyrational.LetmeexplainwhatImean:thespecialtheoryofrelativitywasstrange,irrational,seenagainstthebackgroundof whatwasknownbefore.Therewasnosimplewaytodevelopitthroughanalogieswithpreviousideas.Fermiprobablywouldnothavetriedtodevelopsucha revolution.

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Ithinkhehadasupremesenseoftheimportant.Hedidnotdisdainworkontheso­calledsmallerproblems;atthesametime,hekeptinmindtheorderofimportance ofthingsinphysics.Thisqualityismorevitalinphysicsthaninmathematics,whichisnotsouniquelytiedto''reality."Strangelyenough,hestartedasamathematician. Someofhisfirstpaperswithveryelegantresultsweredevotedtotheproblemofergodicmotion.Whenhewantedto,hecoulddoallkindsofmathematics.Tomy surprise,onceonawalkhediscussedamathematicalquestionarisingfromstatisticalmechanicswhichJohnOxtobyandIhadsolvedin1941. Fermi'swillpowerwasobvious,eventotheextentofcontrollinghisimpulsivegestures.Inmyopinion,hedeliberatelyavoidedvolatileLatinmannerisms,andperhaps byaconsciousdecisioncontrolledgesticulationsandavoidedexclamations.ButEnricosmiledandlaughedveryreadily. Inallactivities,scientificorotherwise,hehadamixtureofsemi­logicalwhimsicalhumoraboutcommon­sensepointsofview.Whenheplayedtennis,forinstance,if helostfourgamestosix,hewouldsay:"Itdoesnotcountbecausethedifferenceislessthanthesquarerootofthesumofthenumberofgames."(Thisisameasureof purelyrandomfluctuationsinstatistics.) Helovedpoliticaldiscussions,andhelovedtrying—nottooseriously—toforeseethefuture.Hewouldaskpeopleinagrouptowritedownwhattheythoughtwould happen,andputitinasealedenvelopetobeopenedacoupleofmonthslater.Onthewholehewasverypessimisticaboutthelong­rangeoutlookpolitically, concludingthathumanityisstillfoolishandwoulddestroyitselfoneday. Hecouldbealsoquiteatease.IrememberhisItalianinflectionswhenhewouldtauntTellerwithstatementslike:"Edward­ahowcom­atheHungarianshavenot­a inventedanything?"OnceSegré,whowasveryfondoffishingonweekendsinthestreamsoftheLosAlamosmountains,wasexpoundingonthesubtletiesoftheart, sayingthatitwasnoteasytocatchtrout.Enrico,whowasnotafisherman,

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saidwithasmile,"Oh,Isee,Emilio,itisabattleofwits." Inconversationswithfriendsaboutthepersonalitiesofothers,hetriedtobeentirelydetachedandobjective,allowinglittleofpersonalorsubjectiveopinionsor feelingstosurface.Abouthimself,hehadtremendousself­assurance.Heknewthathehadthetouchaswellasluckontopofhissupremecommonsense,enormous mathematicaltechnique,andknowledgeofphysics. Enricowasfondofwalking;severaltimeswewalkedallthewayfromLosAlamosdownthewallsofacanyonandalongastreamtotheBandelierNational Monument.Itwasawalkofsevenoreightmilesduringwhichwehadtocrossthestreammorethanthirtytimes.Thewalklastedseveralhours,andwediscussed manysubjects. Ishouldmentionhereoneofmyownpeculiarities:Idonotlikewalkinguphill.Idon'treallyknowwhy.SomepeopletellmethatItendtogotoofastfromimpatience andgetwindedforthatreason.IdonotmindwalkingonlevelgroundandIactuallyenjoywalkingdownhill.YearsagoIboughtaGermantravelguidebookcalled "OneHundredDownhillWalksintheAlps."Certainlyahumoroustitle. Afterthewar,ononeofthesedownhillexcursionsinFrijolesCanyon,ItoldFermihowinmylastyearofhighschoolIwasreadingpopularaccountsoftheworkof Heisenberg,Schrödinger,andDeBroglieonthenewquantumtheory.IlearnedthatthesolutionoftheSchrödingerequationgiveslevelsofhydrogenatomswitha precisionofsixdecimals.Iwonderedhowsuchanartificiallyabstractedequationcouldworktobetterthanonepartinamillion.Apartialdifferentialequationpulled outofthinair,itseemedtome,despitetheappearancesofderivationbyanalogies.IwasrelatingthistoFermi,andatoncehereplied:"It[theSchrödingerequation] hasnobusinessbeingthatgood,youknow,Stan." HewentontosaythatinthefallheintendedtogiveareallylogicalintroductionandderivationofquantumtheoryinhiscourseattheUniversityofChicago.He apparently

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workedatit,buttoldmethenextsummer,whenhereturnedtoLosAlamos,"No,Ididn'tsucceedtomysatisfactioningivingareallyrationalintroductiontoquantum theory."Itisnotjustaquestionofaxiomsassomenaivepuristmightthink.Thequestioniswhysuchandnootheraxiom?Anyworkingalgorithmcanbeaxiomatized. Howtointroduce,justify,tieuporsimplifytheaxioms,historicallyorconceptually,andhowtobasethemonexperiments—thatistheproblem. VonNeumannandFermiwerereallyquitedifferentinpersonality.JohnnywasperhapsbroaderinhisintereststhanEnrico.Hehadmorespecificallyexpressed interestsinotherfields,certainly,forexample,inancienthistory.Fermididnotshowanygreatinterestinorlikingforthearts.Ineverrememberhimdiscussingmusic, painting,orliterature.Currentaffairs,politics,yes;history,no.VonNeumannwasinterestedinboth.Fermididnotindulgeinquotationsorallusions,Latinor otherwise,althoughhelikedepigrammaticformulationsoccasionally.Buthedidnotdisplayagymnasiumorlycéetypeofeducationortheresultantmentalhabits.His overwhelmingcharacteristicwashisLatinclarity.VonNeumanndidnotconsciouslyinsistonsimplicity;onthecontrary,helikedtoshowclevercomplicationson occasion. Intheirlecturestostudentsorscientificgatherings,theydemonstratedtheirdifferentapproaches.Johnnydidnotmindshowingoffbrilliancyorspecialingenuity;Fermi, onthecontrary,alwaysstrivedfortheutmostsimplicity,andwhenhetalkedeverythingappearedinamostnatural,direct,bright,clearlight.Afterstudentshadgone home,theywereoftenunabletoreconstructFermi'sdazzlinglysimpleexplanationofsomephenomenonorhisdeceptivelysimple­lookingideaonhowtotreata physicalproblemmathematically.Incontrast,vonNeumannshowedtheeffectsofhissojournsatGermanuniversities.Hewasabsolutelydevoidofpomposity,butin hislanguagestructurehecouldbecom­

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plicated,thoughperfectlogicalwaysgaveauniqueinterpretationtohiswords. Theyheldhighopinionsofeachother.IrememberadiscussionofsomehydrodynamicalproblemFermihadbeenthinkingabout.VonNeumannshowedawayto considerit,usingaformalmathematicaltechnique.Fermitoldmelaterwithadmiration,"Heisreallyaprofessional,isn'the!"AsforvonNeumann,healwaystook externalevidencesofsuccessseriously;hewasquiteimpressedbyFermi'sNobelPrize.Healsoappreciatedwistfullyotherpeople'sabilitytogetresultsbyintuitionor seeminglypureluck,especiallybytheapparenteffortlessnessofFermi'sfundamentalphysicsdiscoveries.Afterall,Fermiwasperhapsthelastall­aroundphysicistin thesensethatheknewthetheory,didoriginalworkinmanybranches,andknewwhatexperimentstosuggestandevendohimself;hewasthelasttobegreatbothin theoryandasanexperimenter. NielsBohr,thediscovererofthequantizedelectronorbitsintheatomandagreatpioneerofquantumtheory,wasinLosAlamosforseveralmonths.Hewasnotvery old.Tomeatthirty­five,heseemedancient,eventhoughBohrinhislatefiftieswasveryactiveandenergetic,physicallyaswellasmentally.Hewalked,skied,and hikedintheLosAlamosmountains.Somehowheseemedtheembodimentofwisdom.(Wisdom,perhapsnotgeniusinthesenseofNewtonorEinstein.)Heknew whatnottoattemptandhowmuchcouldbedonewithoutmathematics,whichhelefttoothers.ThisenormouswisdomiswhatIlikedabouthim. Departingfromhisusualcautionaboutexpressingopinionsaboutotherpeople,FermiremarkedoncethatwhenBohrtalkedhesometimesgavetheimpressionofa Catholicpriestcelebratingmass.Itwasaniconoclasticstatement,sincesomanyphysicistsarestillunderthespellofBohr. Hehadhisownkindofgeniusthatmadehimagreatphysicistbut,tomymind,someofhisstudentswerealmost

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benightedbyhiscomplementarityphilosophyof"onecansaythis,butontheotherhandonecan..."or"onecannotsaysharplywhatthismeans."Peoplewithouthis greatsenseandintuitivewisdomwereledastrayandlosttheprecisionandsharpnessoftheirintellectualorscientificapproach,inmyopinion.Buthestillhasmany admirers.VictorF.Weisskopfisone. Itseemstomethatasaphilosophicalguideline,complementarityisessentiallynegative.Itcanonlyconsole.Whetheritcanbepositivelyusefulotherthanin philosophicalconsolationsisaquestionwhichtroublesme. Bohr'sspeechwasverydifficulttounderstand,andanecdotesabouthimabound.Mostofthetimeitwasimpossibletogethisexactwords.Onedayayoung physicist,RubyScherr,wascalledonthepublicaddresssystem.HereIshouldexplainthateverydayatperiodicintervalsthehallsofthelaboratoryresoundedwith announcementsandrequests,themostfrequentofwhichwasacallforJ.J.Gutierrez,whowasasupplyfactotumandjackofalltrades.Othercallswererequestsfor thereturnofsuchandsuchaninstrument,oreventheSears,Roebuckcatalogue.OnedayamongotherannouncementscameoneaskingRubyScherrtopleasegoto NicholasBaker'soffice.(NicholasBakerwasthepseudonymofBohrforsecurityreasons;Fermi'swasFarmer.)AsRubyScherrtellsthestory,hewenttotheoffice, sawseveralphysicistssittingaroundandobviouslylisteningtoapresentationbyBohr.Bohrstopped,mumbledafewincomprehensiblesentencesinthedirectionof Scherr,andsuddenlyendedwithacrystal­clearthreewords:"Guesshowmuch?"Scherr,whohadnotunderstoodawordofthequestion,blushedwith embarrassment,shookhisheadshylyandremainedsilent.AfteramomentBohragaininaclearvoicesaid"1041."Whereuponeverybodylaughed.Tothisday,Scherr doesnotknowwhatitwasallabout. AnotherBohrstoryillustratestheabsentmindednessofscientists:itwaswellknownthroughoutLosAlamosthatNicholasBakerwasNielsBohr;nevertheless,histrue name

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wasneversupposedtobementionedinpublic.Atonecolloquium,Weisskopfreferredto"thewell­knownBohrprinciple.""Ohexcuseme,"hefumbled,"the NicholasBakerprinciple!"Generallaughtergreetedthissecuritybreach. Notallofuswereundulysecurityconscious.Everyscientist,oldoryoung,hadinhisofficeasafewheresecretdocumentshadtobekept.Indeed,theProjectmust havehadmoresafesthanallthebanksinNewYork.OnceinBohr'sofficeIwatchedhimstruggletoopenhissafe.Thesafescouldbeopenedwitharathersimple combinationofthreetwo­digitnumbers.Hetriedandtriedforalongtime,finallysucceeding.Hepulledoutthedrawerandexclaimeddelightedly:"IbelieveIhave doneenoughfortheday."ThestorythatDickFeynmancouldopensafeswhosecombinationshadbeenforgottenbytheirownersistrue.Heapparentlylistenedto theclicksofthetumblersandsometimesheguessedwhichcombinationsofdigitsofnumberslike oreinmathematics,orc,thevelocityoflight,orh,Planck's constant,hadbeenselectedbytheownersforthecombinations. Onethingthatrelievedtherepetitionandalternationofwork,intellectualdiscussions,eveninggatherings,socialfamilyvisitsanddinnerparties,waswhenagroupofus wouldplaypokeraboutonceaweek.ThegroupincludedMetropolis,Davis,Calkin,Flanders,Langer,Long,Konopinski,vonNeumann(whenhewasintown), Kistiakowskisometimes,Teller,andothers.Weplayedforsmallstakes;thenaïvetéofthegameandthefrivolousdiscussionslacedwithearthyexclamationsand roughlanguageprovidedabathofrefreshingfoolishnessfromtheveryseriousandimportantbusinessthatwastheraisond'êtreofLosAlamos. Inplayingsuchagame,unlessyouarevitallyinterestedinthegameitself,andnotmerelyinitsrelaxingqualities,youwillnotdowell.VonNeumann,Teller,andI wouldthinkaboutcompletelyunrelatedsubjectsduringthebiddingorbetting;consequently,moreoftenthannotwewerethelosers.Metropolisoncedescribedwhat atriumphitwas

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towintendollarsfromJohnvonNeumann,authorofafamoustreatiseongametheory.Hethenboughthisbookforfivedollarsandpastedtheotherfiveinsidethe coverasasymbolofhisvictory.Itmaynotbecleartonon­scientistsornon­mathematiciansthatonecandotheoreticalworkinone'sheadandpursueitquite intenselywhileliterallycarryingonsomeothermoreprosaicactivity. TheTrinitytest,Hiroshima,V­JDay,andthestoryofLosAlamosexplodedovertheworldalmostsimultaneouslywiththeA­Bomb.Publicityoverthesecretwartime Projectfilledthenewspapersanditsadministrativeheadswerethrowninthelimelight.InonenewspaperinterviewoutofmanypublishedthedayafterHiroshima,E. O.Lawrence"modestlyadmitted,"accordingtotheinterviewer,"thathemorethananyoneelsewasresponsiblefortheatomicbomb."Similarstatementsbyand aboutothersfilledthemedia.OppenheimerwasreportedtohavedescribedhisfeelingsaftertheunearthlylightoftheinitialflashoftheTrinityexperimentbyquoting fromtheHinduepic,theBhagavadGita:"ItflashedtomymindthatIhadbecomethePrinceofDarkness,thedestroyerofUniverses." WhatistrueisthatasIwasreadingthisiteminanewspaper,somethingelseflashedthroughmymind,astoryofa"pension"inBerlinbeforethewar.Itoldit immediatelytoJohnny,whowaseatingdinnerinourhouse.TheBerlinboardersweresittingaroundatablefordinneranddisheswerepassedforeachpersontohelp himselftohisshare.Onemanwastakingmostoftheasparagusthatwasontheplatter.Whereuponanothermanstoodupshylyandsaid:"Excuseme,Mr.Goldberg, wealsolikeasparagus!"Andtheexpression"asparagus"becameitcodewordinourprivateconversationsfortryingtoobtainanundulylargeshareofcreditfor scientificworkoranyotheraccomplishmentofajointorgroupcharacter.Johnnylovedthisstorysomuchthatinourhumorousconversationsweplayedon developingthetheme.Wewouldplantowriteatwenty­volumetreatiseon"AsparageticsthroughtheAges.''Johnnywould

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do"DieAsparageticsimAltertum"andIthefinalvolume"RückblickundAusblick"inthemannerofheavyGermanscholarship.Later,CarsonMarkputhisown stamponthesejokesbycomposingasong,"Oh,HowILoveAsparagus,"tothetuneofacurrentpopularsong. ButlevitieslikethesecouldhardlyalleviatethegeneralfeelingofforebodinguponenteringintotheeraofhistorythatwouldbecalledtheAtomicAge.Thewarwas over,theworldandthenationhadtoreorganizethemselves.Lifewouldneverbethesame.

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Chapter9 SouthernCalifornia 1945–1946 Thewarwasoverandtheworldwasemergingfromtheashes.ManypeopleleftLosAlamos,eithertoreturntotheirformeruniversitieslikeHansBethe,ortogoto newacademicpositionslikeWeisskopftoMITorTellertoChicago.Thegovernmenthadnotreachedanydecisionyetaboutthefateofthewartimelaboratory. TheUniversityofChicagotookstepstostartagreatnewcenterfornuclearphysics,withFermi,TellerandseveralothersfromtheManhattanDistrictProject.Von Neumann,betterthananyoneelse,itseemstome,arguedthatasaresultoftherolesciencehadplayedinthewinningofthewar,thepost­waracademicworldwould notberecognizableinpre­1939terms. Onthepurelypersonalplane,Ihadnoevidencethatanymemberofmyimmediatefamilyhadsurvived(twocousinsdidreappearmanyyearslater,oneinFrance,the otherinIsrael).Françoisehadlosthermotherintheconcentration

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campofAuschwitz.WewerebothAmericancitizensnow,theUnitedStateswasourcountry,andtheideaofreturningtoEuropeneverenteredourheads.Butthe questionofwhatjobtoreturntofromwarworkwasverymuchonourminds. IhadsomecorrespondencewithLanger,whowasthenthechairman,aboutreturningtoMadison.Hewasveryhonestandopen,andhetoldmewithadmirable franknesswhenIinquiredaboutmychancesforpromotionandtenure:"Noreasontobeataroundthebush,wereyounotaforeigner,itwouldbemucheasierand yourcareerwoulddevelopfaster."SoitseemedthatmychancesinWisconsinwerenotverygood,andIlookedelsewhere.Elsewherecameintheformofaletter fromanoldMadisonfriend,DonaldHyers,whohadbecomeaprofessorattheUniversityofSouthernCaliforniainLosAngeles.Hyerswaswellestablishedthere, andheaskedwhetherIwouldbeinterestedinjoiningthefacultyasanassociateprofessoratasalarysomewhathigherthantheoneinMadison.Theuniversitywas small,notverystrongacademically,andcertainlynotaveryprestigiousplace,buttheprofessorsthere,hesaid,wereengagedinvigorousattemptsatimprovingthe academicstandingoftheinstitution.Heinvitedmeforavisit,andIflewtoLosAngelesinAugustof1945. ThiswasthefirsttimeIsawthatcity,anditgavemeaverystrangeimpression.ItwasadifferentworldfromanyIhadknown,climatically,architecturally,and otherwise.ImentionedthisjobpossibilitytoJohnny,andalthoughhewasrathersurprisedatmyinterestinthisrathermodestopportunity,hedidnotreactnegatively. Histendencywastogoalong.IdidnotseemuchsenseinmarkingtimeinLosAlamosafterthewar,soIacceptedtheUSCoffer. InearlySeptemberof1945,IwenttoLosAngelestolookforhousingandtoprepareourmovefromLosAlamos.Intheimmediatepostwarperiod,thehousing situationinLosAngeleswascritical.Sincewedidnotownacar,wewererestrictedtosearchingforahouseinthevicinityofthe

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University.IusedtosaythatanytwopointsinLosAngeleswereatleastanhour'sdriveapart,a"discrete"topologicalspace.Imanagedtosubletforonesemestera typicalsmallLosAngeleshouseonamodeststreetlinedwithspindlypalmtrees.Tomeitseemedadequate,butitappearedrathermiserabletoFrançoise. Nevertheless,wesettledtheretemporarilyforlackofanythingbetter.Inoticedthatinourvariousmovesfromonehabitattothenextallourmaterialpossessions, clothes,books,furnishingshadawayofdiminishingintransit.Iusedtosaythattheydwindledto1/e,inanalogytotheenergylossesofparticlesintransitthrough"one meanfreepath." Forthesecondsemesterofthatacademicyear(1945–46),HalandHattievonBreton,goodfriendsoftheHawkinses,invitedustostayintheirsummercottageon BalboaIslandacrossfromNewportBeach.Itwasonthewater,beautifulandcomfortable—awonderfulchangefromtheuniversityneighborhoodbutalittletoofar formetocommutedaily—soduringtheweekIlivedinahotelnearthecampusandwenthometotheislandonweekends.FrançoiseremainedonBalboawithour babydaughterClaire,whohadbeenborninLosAlamostheyearbefore. AtUSCIfoundtheacademicatmospheresomewhatrestricted,ratheranticlimacticaftertheintensityandthehighlevelofscienceatLosAlamos.Everyonewasfullof goodwill,evenifnotterriblyinterestedin"research."The"teachingload"towhichIwasreluctantlyreturningwasnottooheavy.Allinall,thingslookedpromisinghad itnotbeenforaviolentillnesswhichstruckmesuddenly.IhadreturnedtoLosAngelesfromamathematicsmeetinginChicagowithamiserablecold.Itwasastormy day;onthewalkfromthebustothehouseinBalboatheviolentwindsalmostchokedme.ThatsamenightIdevelopedafantasticheadache.NeverinmylifehadI experiencedaheadacheofanykind;thiswasanewfeelingaltogether—themostseverepainIhadeverendured,all­pervadingandconnectedwithasensationof numbnesscreepingupfromthebreast

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bonetothechin.IrememberedsuddenlyPlato'sdescriptionofSocratesafterhewasgiventhehemlockinprison;thejailormadehimwalkandtoldhimthatwhenthe feelingofnumbnessstartinginthelegsreachedhisheadhewoulddie. Françoisehaddifficultyinfindingadoctorwhowouldcometotheislandinthemiddleofthenight.Theonewhofinallycamecouldnotfindanythingvisiblywrongand gavemeashotofmorphinetoalleviatetheexcruciatingpain.ThenextmorningIfeltalmostnormalbutwithalingeringfeelingoflassitudeandaninabilitytoexpress myselfclearly,whichcameandwent.Nevertheless,IreturnedtoLosAngelesandgavemylecturesattheuniversity.Thefollowingnighttheviolentheadache reappeared.WhenItriedtotelephoneFrançoisefrommyhotelroom,Inoticedthatmyspeechwasconfused,thatIwasbarelyabletoformwords.Itriedtotalk aroundtheexpressionswhichwouldnotcomeoutandformequivalentones,butitwasmostlyameaninglessmumble—amostfrighteningexperience.Greatlyalarmed bymyincoherentphonecall(Idon'tknowhowImanagedtorememberthephonenumberathome),FrançoisecalledthevonBretonsandaskedthemtosenda doctortoseeme.Infact,twodoctorsappeared.Perplexedbymysymptomsthatcameandwent,theytookmetoCedarsofLebanonhospital.Asevereattackof braintroublesbegan,whichwastobeoneofthemostshatteringexperiencesofmylife.Bytheway,manyoftherecollectionsofwhatprecededmyoperationare hazy.ThankstowhatFrançoisetoldmelaterIwasabletoputittogether. ForseveraldaysIunderwentvarioustests—encephalograms,spinaltaps,andthelike.Theencephalogramwaspeculiar.Thedoctorssuspectedatumor,whichcould bebenignormalignant.Dr.Rainey,aneurosurgeonpupilofCushing,wascalledinandanoperationwasplannedforthefollowingday.OfallthisIknewnothing,of course.Irememberonlytryingtodistractthenurse'sattentionbytellinghertolookoutofthewindowsoIcouldreadmychart.I

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sawtheresomealarmingnotationaboutC­3whichIsuspectedtomeanthethirdconvolutionofthebrain.ThroughallthisIwasovercomebyanintensefearand begantothinkIwasgoingtodie.Iconsideredmychancesofsurvivingtobelessthanhalf.Theaphasiawasstillpresent;muchofthetimewhenItriedtospeakI utteredmeaninglessnoises.IdonotknowwhynoonethoughtofascertainingwhetherIcouldwriteinsteadofspeak. Françoise,alertedbythevonBretons,rushedallthewaybackfromBalboabytaxiandarrivedonthescenejustasIwasbeginningtovomitbile,turninggreenand losingconsciousness.ShefearedIwasdyingandmadeafrantictelephonecalltothesurgeon,whodecidedtheoperationshouldbeperformedimmediately.This probablysavedmylife;theemergencyoperationrelievedtheseverepressureonmybrainwhichwascausingallthetrouble.Irememberthatinmysemi­conscious statemyheadwasbeingshavedbyabarber(hehappenedtobeaPole)whosaidafewwordsinPolish,towhichItriedtoreply.Irememberalsoreturningto consciousnessbrieflyinapre­operatingroomandwonderingwhetherIwasalreadyinthemorgue.Ialsorememberhearingthenoiseofadrill.Thiswasatrue sensationasitturnedout,forthedoctorsdrilledaholeinmyskulltotakesomelast­minuteX­rays.Thesurgeonperformedatrepanationnotknowingexactlywhere orwhattolookfor.Hedidnotfindatumor,butdidfindanacutestateofinflammationofthebrain.HetoldFrançoisethatmybrainwasbrightpinkinsteadofthe usualgray.Theseweretheearlydaysofpenicillin,whichtheyappliedliberally.A"window"wasleftonthebraintorelievethepressurewhichwascausingthe alarmingsymptoms. Iremainedinapost­operativecomaforseveraldays.WhenIfinallywokeup,Ifeltnotonlybetter,butpositivelyeuphoric.Thedoctorspronouncedmesaved,even thoughtheytoldFrançoisetoobservemeforanysignsofchangesofpersonalityorrecurrenceofthetroubleswhichwouldhavespelledbraindamageorthepresence ofahidden

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growth.Iunderwentmoretestsandexaminations,andtheillnesswastentativelydiagnosedasakindofvirusencephalitis.Butthedisquietudeaboutthestateofmy mentalfacultiesremainedwithmeforalongtime,eventhoughIrecoveredspeechcompletely. Onemorningthesurgeonaskedmewhat13plus8were.ThefactthatheaskedsuchaquestionembarrassedmesomuchthatIjustshookmyhead.Thenheasked whatthesquarerootoftwentywas,andIreplied:about4.4.Hekeptsilent,thenIasked,"Isn'tit?"IrememberDr.Raineylaughing,visiblyrelieved,andsaying,"I don'tknow."AnothertimeIwasfeelingmyheavilybandagedhead,andthedoctorchidedmesayingthebacteriacouldinfecttheincision.IshowedhimIwas touchingadifferentplace.ThenIrememberedthenotionofameanfreepathofneutronsandaskedhimifheknewwhatthemeanfreepathofbacteriawas.Instead ofanswering,hetoldmeanunprintablejokeaboutamansittingonacountrytoiletandhowthebacterialeapedfromthesplashingwater.Thenursesseemedtolike meandofferedallkindsofmassagesandbackrubsandspecialdiets,whichhelpedmymoralemorethanmyphysicalcondition(whichwassurprisinglygood). Manyfriendscametovisitme.JackCalkin,whowasonleaveonCatalinaIsland,appearedseveraltimesatthehospital.SodidcolleaguesfromtheUniversity.I rememberthemathematicianAristotleDimitriosMichael.HetalkedsoagitatedlythatIfelloutofbedlisteningtohim.Thisscaredhimverymuch.ButImanagedto scramblebackeventhoughIwasstillslightlynumbononeside.NickMetropoliscameallthewayfromLosAlamos.Hisvisitcheeredmegreatly.Ifoundoutthatthe securitypeopleinLosAlamoshadbeenworriedthatinmyunconsciousorsemi­consciousstatesImighthaverevealedsomeatomicsecrets.Therewasalsosome questionastowhetherthisillness(whichwasneverproperlydiagnosed)mighthavebeencausedbyatomicradiation.Butinmycasethiswashighlyimprobable,forI hadneverbeenclosetoradioactivematerial,having

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workedonlywithpencilandpaper.Universityofficialsvisitedme,too.TheyseemedconcernedaboutmyabilitytoresumemyteachingdutiesafterIgotwell.People wereacutelyconcernedaboutmymentalfaculties,wonderingwhethertheywouldreturninfull.Iworriedmyselfagooddealaboutthat,too;wouldmyabilitytothink returninitsentiretyorwouldthisillnessleavemementallyimpaired?Obviouslyinmyprofession,completerestorationofmemorywasofparamountimportance.I wasquitefrightened,butinmyself­analysisInoticedthatIcouldimagineevengreaterstatesofpanic.Logicalthoughtprocessesareverymuchdisturbedbyfright. Perhapsitisnature'swayofblockingtheprocessintimesofdangertoallowinstincttotakeover.Butitseemstomethatmereinstincts,whichresideinnervesandin muscle"programming,"arenolongersufficienttocopewiththecomplicatedsituationsfacingmodernman;somesortofreasoningabilityisstillneededinthefaceof mostdangeroussituations. Iregainedmystrengthandfacultiesgraduallyandwasallowedtoleavethehospitalafterafewweeks.Iobtainedaleaveofabsencefromtheuniversity. Irememberbeingdischargedfromthehospital.AsIwaspreparingtoleave,fullydressedforthefirsttime,standinginthecorridorwithFrançoise,Erdösappearedat theendofthehall.Hedidnotexpecttoseemeup,andheexclaimed:"Stan,Iamsogladtoseeyouarealive.IthoughtyouweregoingtodieandthatIwouldhave towriteyourobituaryandourjointpapers."Iwasveryflatteredbyhispleasureatseeingmealive,butalsoveryfrightenedtorealizethatmyfriendshadbeenonthe brinkofgivingmeupfordead. ErdöshadasuitcasewithhimandwasjustleavingafteravisittoSouthernCalifornia.Hehadnoimmediatecommitmentsaheadandsaid,"Youaregoinghome? Good,Icangowithyou."SoweinvitedhimtocomewithustoBalboaandstayawhile.Theprospectofhiscompanydelightedme.Françoisewassomewhatmore dubious,fearingthatit

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wouldtiremetoomuchduringtheearlypartofmyconvalescence. AmathematicalcolleaguefromUSCdroveusallbacktothevonBretons'houseonBalboaIsland.Physically,Iwasstillveryweakandmyheadhadnotyethealed.I waswearingaskullcaptoprotecttheincisionuntilmyhairgrewback,Irememberhavingdifficultywalkingaroundtheblockthefirstfewdays,butgraduallymy strengthreturned,andsoonIwaswalkingamileeachdayonthebeach. Inthecaronthewayhomefromthehospital,Erdösplungedimmediatelyintoamathematicalconversation.Imadesomeremarks,heaskedmeaboutsomeproblem, Imadeacomment,andhesaid:"Stan,youarejustlikebefore."Thesewerereassuringwords,forIwasstillexaminingmyownmindtryingtofindoutwhatImight havelostfrommymemory.Paradoxically,onecanperhapsrealizewhattopicsonehasforgotten.NosoonerhadwearrivedthanErdösproposedagameofchess. AgainIhadmixedfeelings:ononehandIwantedtotry;ontheother,IwasafraidtoincaseIhadforgottentherulesofthegameandthemovesofthepieces.Wesat downtoplay.IhadplayedalotofchessinPolandandhadmorepracticethanhehad,andImanagedtowinthegame.Butthefeelingofelationthatfollowedwas immediatelytemperedbythethoughtthatperhapsPaulhadletmewinonpurpose.Heproposedasecondgame.Iagreed,althoughIfelttired,andwonagain. WhereuponitwasErdöswhosaid,"Letusstop,Iamtired."Irealizedfromthewayhesaiditthathehadplayedinearnest. Inthedaysthatfollowedwehadmoreandmoremathematicaldiscussionsandlongerandlongerwalksonthebeach.Oncehestoppedtocaressasweetlittlechild andsaidinhisspeciallanguage:"Look,Stan!Whataniceepsilon."Averybeautifulyoungwoman,obviouslythechild'smother,satnearby,soIreplied,"Butlookat thecapitalepsilon."Thismadehimblushwithembarrassment.Inthose

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dayshewasveryfondofusingexpressionslikeSF(supremefascist)forGod,Joe(Stalin)forRussia,Sam(UncleSam)fortheUnitedStates.Thesewereforhim objectsofoccasionalscorn. Graduallymyself­confidencereturned,buteverytimeanewsituationoccurredinwhichIcouldtestmyreturningpowersofthought,Iwasbesetbydoubtsand worries.Forexample,IreceivedaletterfromtheMathematicalSocietyaskingmeifIwouldwritefortheBulletinanobituaryarticleonBanach,whohaddiedinthe fallof1945.Thisagaingavemereasontoponder.Itseemedalittlemacabreafterhavingbarelyescapeddeathmyselftowriteaboutanother'sdemise.ButIdidit frommemory,nothavingalibraryaround,andsentinmyarticlewithapprehension,wonderingifwhatIhadwrittenwasweakorevennonsensical.Theeditorsreplied thatthearticlewouldappearinthenextissue.YetmysatisfactionandreliefwereagainfollowedbydoubtforIknewthatallkindsofarticleswereprinted,andIdid nothavesuchahighopinionofmanyofthem.Istillfeltunsurethatmythinkingprocesswasunimpaired. Normallyprimitiveor''elementary"thoughtsarereactionstoorconsequencesofexternalstimuli.Butwhenonestartsthinkingaboutthinkinginasequence,Ibelieve thebrainplaysagame—somepartsprovidingthestimuli,theothersthereactions,andsoon.Itisreallyamulti­persongame,butconsciouslytheappearanceisofa one­dimensional,purelytemporalsequence.Oneisonlyconsciouslyawareofsomethinginthebrainwhichactsasasummarizerortotalizeroftheprocessgoingon andthatprobablyconsistsofmanypartsactingsimultaneouslyoneachother.Clearlyonlytheone­dimensionalchainofsyllogismswhichconstitutesthinkingcanbe communicatedverballyorwrittendown.Poincaré(andlaterPolya)triedtoanalyzethethoughtprocess.WhenIrememberamathematicalproof,itseemstomethatI rememberonlysalientpoints,markers,asitwere,ofpleasureordifficulty.Whatiseasyiseasilypassedoverbecauseitcanbereconstitutedlogicallywith

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ease.If,ontheotherhand,Iwanttodosomethingnewororiginal,thenitisnolongeraquestionofsyllogismchains.WhenIwasaboyIfeltthattheroleofrhymein poetrywastocompelonetofindtheunobviousbecauseofthenecessityoffindingawordwhichrhymes.Thisforcesnovelassociationsandalmostguarantees deviationsfromroutinechainsortrainsofthought.Itbecomesparadoxicallyasortofautomaticmechanismoforiginality.Iamprettysurethis"habit"oforiginality existsinmathematicalresearch,andIcanpointtothosewhohaveit.Thisprocessofcreationis,ofcourse,notunderstoodnordescribedwellenoughatpresent. Whatpeoplethinkofasinspirationorilluminationisreallytheresultofmuchsubconsciousworkandassociationthroughchannelsinthebrainofwhichoneisnot awareatall. Itseemstomethatgoodmemory—atleastformathematiciansandphysicists—formsalargepartoftheirtalent.Andwhatwecalltalentorperhapsgeniusitself dependstoalargeextentontheabilitytouseone'smemoryproperlytofindtheanalogies,past,presentandfuture,which,asBanachsaid,areessentialtothe developmentofnewideas. Icontinuetospeculateonthenatureofmemoryandhowitisbuiltandorganized.Althoughonedoesnotknowmuchatpresentaboutitsphysiologicaloranatomical basis,whatgivesapartialhintishowonetriestorememberthingswhichonehastemporarilyforgotten.Thereareseveraltheoriesaboutthephysicalaspectsof memory.Someneurologistsorbiologistssaythatitconsistsperhapsofpermanentlyrenewedcurrentsinthebrain,muchasthefirstcomputermemorieswerebuilt withsoundwavesinamercurytank.OtherssaythatitresidesinchemicalchangesofRNAmolecules.Butwhateveritsmechanism,animportantthingistounderstand theaccesstoourmemory. Experimentsseemtoindicatethatthememoryiscompleteinthesensethateverythingweexperienceorthinkaboutisstored.Itisonlytheconsciousaccesstoitthat is

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partialandvariesfrompersontoperson.Someexperimentshaveshownthatbytouchingacertainspotinthebrainasubjectwillseemtorecalloreven"feel"a situationthathappenedinthepast—suchasbeingataconcertandactuallyhearingacertainmelody. Howismemorygraduallybuiltupduringone'sconsciousorevenunconsciouslifeandthought?Myguessisthateverythingweexperienceisclassifiedandregistered onverymanyparallelchannelsindifferentlocations,muchasthevisualimpressionsthataretheresultofmanyimpulsesondifferentconesandrods.Allthesepictures aretransmittedtogetherwithconnectedimpressionsfromothersenses.Eachsuchgroupisstoredindependently,probablyinagreatnumberofplacesunderheadings relevanttothevariouscategories,sothatinthevisualbrainthereisapicture,andtogetherwiththepicturesomethingaboutthetime,orthesource,ortheword,orthe sound,inabranchingtreewhichmusthaveadditionallyanumberofconnectingloops.Otherwiseonecouldnotconsciouslytryandsometimessucceedin rememberingaforgottenname.Inacomputingmachine,oncetheaddressofthepositionofaniteminthememoryislost,thereisnowaytogetatit.Thefactthatwe succeed,atleastonoccasion,meansthatatleastonememberofthe"searchparty"hashitaplacewhereanelementofthegroupisstored.Thusitiscommontorecall alastnameoncethefirstnamehasbeenrecalled. ThenIthought,howaboutsmell?Smellissomethingwesense;itisnotrelatedtoanysoundorpicture.Wedonotknowhowtocallit.Ithasnovisualimpacteither. Doesthiscontradictmyguessesaboutsimultaneousstorageandconnections?ThenIrememberedthefamousincidentrelatedbyProustofthesmellandtasteofthe "madeleine"(littlecake).Therearemanydescriptionsintheliteratureofcaseswhereasmellpreviouslyexperiencedandfeltsuddenlybringsbackalong­forgotten occasionwhenitwasfirstassociatedwithaplace,oraperson,manyyearsbefore.So,perhapsonthecontrary,thisisanotherindication.

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Thisfeelingofanalogyorassociationisnecessarytoplacethesetofimpressionscorrectlyonthesuitableendpointsofasequenceofbranchesofatree.Andperhaps thisishowpeopledifferfromeachotherintheirmemories.Insome,moreoftheseanalogiesarefelt,stored,andbetterconnected.Suchanalogiescanbeofan extremelyabstractnature.Icanconceivethataconcretepicture,avisualsequenceofdotsanddashes,maybringbackanabstractthought,whichapparentlyina mysteriouscodinghadsomethingincommonwithit.Somepartofwhatiscalledmathematicaltalentmaydependontheabilitytoseesuchanalogies. Itissaidthatseventy­fivepercentofushaveadominantvisualmemory,twenty­fivepercentanauditoryone.Asforme,mineisquitevisual.WhenIthinkabout mathematicalideas,Iseetheabstractnotionsinsymbolicpictures.Theyarevisualassemblages,forexample,aschematizedpictureofactualsetsofpointsonaplane. Inreadingastatementlike"aninfinityofspheresoraninfinityofsets,"Iimagineapicturewithsuchalmostrealobjects,gettingsmaller,vanishingonsomehorizon. Itispossiblethathumanthoughtcodesthingsnotintermsofwordsorsyllogismsorsigns,formostpeoplethinkpictorially,notverbally.Thereisawayofwriting abstractideasinakindofshorthandwhichisalmostorthogonaltotheusualwaysinwhichwecommunicatewitheachotherbymeansofthespokenorwrittenword. Onemaycallthisa"visualalgorithm." Theprocessoflogicitselfworkinginternallyinthebrainmaybemoreanalogoustoasuccessionofoperationswithsymbolicpictures,asortofabstractanalogueof theChinesealphabetorsomeMayandescriptionofevents—exceptthattheelementsarenotmerelywordsbutmorelikesentencesorwholestorieswithlinkages betweenthemformingasortofmeta­orsuper­logicwithitsownrules. Forme,someofthemostinterestingpassagesabouttheconnectionsbetweentheproblemoftime,asinvolvedin

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thememory,andthephysicalorevenmathematicalmeaningofit,whetheritisclassicalorrelativistic,werewritten,notbyaphysicistoraneurologistoraprofessional psychologist,butbyVladimirNabokovinhisbookAda.SomeutterancesbyEinsteinhimself,asquotedinhisbiographies,showthegreatphysicist'swonderatwhat livingintimemeans,sinceweexperienceonlythepresent.But,inreality,weconsistofpermanentandimmutableworldlinesinfourdimensions. Withsuchthoughtsandworriesaboutthethinkingprocess,Iwasrecoveringmyphysicalstrengthduringthisperiodofconvalescence.Whatcomfortedmethemost wasthereceiptofaninvitationtoattendasecretconferenceinLosAlamosinlateApril.Thisbecameformeatruesignofconfidenceinmymentalrecovery.Icould notbetoldonthetelephoneorbyletterwhattheconferencewasabout.Secrecywasmostintenseatthattime,butIguessedcorrectlythatitwouldbedevotedtothe problemsofthermonuclearbombs. Theconferencelastedseveraldays.Manyfriendswerepresent.Somehadbeendirectlyinvolved,likeFrankel,Metropolis,Teller,andmyself;otherswere consultants,likevonNeumann.Fermiwasabsent.Thediscussionswereactiveandinquisitive.TheybeganwithapresentationbyFraenkelofsomecalculationsonthe workinitiatedbyTellerduringthewar.Theywerenotdetailedorcompleteenoughandrequiredworkoncomputers(nottheMANIACsbutothermachinesin operationattheAberdeenProvingGrounds).Thesewerethefirstproblemsattackedthatway. Thepromisingfeaturesoftheplanwerenoticedandtosomeextentconfirmed,butthereremainedgreatquestionsabouttheinitiationoftheprocessand,once initiated,aboutitssuccessfulcontinuation. (AllthiswastohavegreatimportanceinalaterlawsuitbetweenSperryRandandHoneywelloverthevalidityofpatentsinvolvingcomputers.Theclaimwasthat computers

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werealreadyinthepublicdomainthenbecausethegovernmentoftheUnitedStatesusedthemandthereforethepatentsgrantedlaterwereinvalid.Iwasoneofmany whowerecalledtotestifyonthisin1971.) IparticipatedinalltheLosAlamosmeetings.Theylastedforhours,morningsandafternoons,andInoticedwithpleasurethatIwasnotundulytired. IremembertellingJohnnyaboutmyillness."Iwasgivenupfordead,"Isaid,"andthoughtmyselfthatIwasalreadydead,exceptforasetofmeasurezero."This purelymathematicaljokeamusedhim,helaughedandasked,''Whatmeasure?" EdwardTellerandJohnnywereoftentogether,andIjoinedtheminprivatetalks. Inoneconversationtheydiscussedthepossibilityofinfluencingtheweather.Theyhadinmindglobalchanges,whileIproposedmorelocalinterventions.Forexample, IrememberaskingJohnnywhetherhurricanescouldnotbediverted,attenuated,ordispersedwithnuclearexplosions.Iwasn'tthinkingofapointsource,whichis symmetrical,butseveralexplosionsinaline.Ireasonedthattheviolenceandenormousenergyofahurricaneliesontopofamassofair(theweather)whichitself movesgentlyandslowly.Iwonderedifonecouldnot,eveneversoslightly,changeitscourseintimeandintrajectoryontheslowmovingoverallweather,thusmaking itavoidpopulatedareas.Thereare,ofcourse,manyquestionsandobjectionsaboutsuchanundertaking.Oneofthenecessaryconditionswouldbetomakedetailed computationsonthecourseofthemotionoftheairmasses,calculationswhichdonotexistevennow.ThroughtheyearsJohnnyandIoccasionallytalkedaboutthis withexpertsinhydrodynamicsandmeteorology. Theconferenceover,IreturnedtoLosAngeles.Uponalightingfromtheplane,twoFBIagentsapproachedme,showedtheiridentificationandaskedforpermission tosearchmyluggage.AcopyoftheverysecretMetropolisandFrankelreportwasmissing,andtheywonderedifImight

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havetakenitbymistake.Wesearched,butIdidnothaveit.LaterIlearnedthateverybodywhohadattendedtheconferencehadbeencontacted.Theauthorities wereverynervous,forthiswaspotentiallyofgraveconsequence.ThemissingdocumentreappearedmuchlateramongsomeofTeller'spapersinaLosAlamossafe. ThetimewasrapidlyapproachingwhenIcouldresumeteaching,butIwasdevelopingstronglynegativefeelingsaboutLosAngeles.RidesthroughthestreetswhereI hadbeendriveninanambulanceremindedmeofmyrecentillness.MyfeelingstowardtheUniversitywerecoloredbythis,aswell,andIwasdissatisfied.Ifelt impatientlythatitwasnotchangingquicklyenoughfromaglorifiedhighschoolintoagenuineinstitutionofhigherlearning.Ihaddisagreementswithadeanabout buildinguptheacademiclevelandincreasingthestaff.Iwastoldhejokedthathealmosthadaheartattackeverytimehesawme,evenfromadistance,soafraidwas hethatIwasbringinghimnewproposalsforexpansion! ThebestpartoftheUniversitywastheHancockLibrary.Ithadanimpressivebuildingandsomegoodbooks—butthebuildingwasbetterthanthecollectioninside. TheUniversityhadjustacquiredanoldmunicipallibraryfromBoston,andwhenIlearnedwhatitcontained,Icomparedittoapricelesscollectionofhundredyearold SearsRoebuckcatalogues.Thissarcasticremarkprobablydidnotenhancemypopularity. EventhoughIhadfriendslikeDonaldHyers,andsomenewacquaintancesamongmathematicians,physicists,andchemists,withthisgrowingdisenchantmentI wantedtoleave.TheLosAngelesexperiencehadnotbeensatisfactory. JustthenIreceivedatelegraminvitingmetoreturntoLosAlamosinabetterpositionandatahighersalary.ItwassignedbyBobRichtmyerandNickMetropolis. Richtmyerhadbecomeheadofthetheoreticaldivision. ThisoffertoreturntoLosAlamostoworkamongphysi­

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cistsandliveonceagainintheexhilaratingclimateofNewMexicowasagreatreliefforme.IrepliedimmediatelythatIwasinterestedinprinciple.Whenthetelegram arrivedatthelaboratory,itreadthatIwasinterested"inprincipal."

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Chapter10 BackatLosAlamos 1946–1949 LosAlamoswasataboutthelowestpointinitsexistence.YetonreturningIfoundthattherewereanumberofpeoplewhohaddecidedtostayonandthatthe governmentwantedtokeepthelaboratorygoingandhaveitflourish.Thelaboratorywastocontinuestudiesandthedevelopmentofatomicbombs. Afterthewartherewas,ofcourse,thequestionofpossiblenewwarsandtheweaponryofthefuture.Iwasinfavorofcontinuingstrongarmamentpoliciesifonlynot toruntheriskofbeingovertakenbyothernations.JohnnyandotherswereapprehensiveaboutRussia'sabilitytoobtainortodevelopnuclearbombs,andaboutits intentionstowardsWesternEurope.Hewasquitehawkishatthattime(thewords"hawks"and"doves"werenotyetinuse).Hethoughtalongtheoldhistoricallines ofrivalries,powerstruggles,

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coalitions;hewasforaPaxAmericanamorethansomeofourotherphysicistfriends.Healsoforesawearlythattheessentialmilitaryproblemswouldshiftfromthe bombsthemselvesandtheirsizesandshapestowaystodeliverthem—thatistosay,torocketry. Myownpositionwassortofhalfwaybetweenhimandthephysicistswhohopedtointernationalizenuclearweapons.Ithoughtitwasnaivetoexpectthatthewolves wouldliedownwiththelambsandfeltthatmeaningfulinternationalagreementswouldtakemanyyears.Onecouldnothopeforaninstantchangeinattitudesorin humannatureitself.IdistrustedtheideaoftheAtlanticUnionasthenproposed,feelingthatsomeofthepropagandaforitwastootransparent.Thehegemony disguisedthinlyunderageneralorganizationwouldmerelyraisefearsandnewhystericalreactionsfromtheotherside.However,Ifailedtorealizefullytheimmense importanceofnucleararmamentandtheinfluenceitwouldhaveonthecourseofworldevents.Onebomb,Itoldmyself,wasequaltoathousandplaneraid.YetIdid notrealizethatthepowerofeachsuchbombcouldbestillvastlyincreased,andthatitwaspossibletomanufacturethousandsofthem.Thisrealizationcamelater.I feltnoqualmsaboutreturningtothelaboratorytocontributetofurtherstudiesofthedevelopmentofatomicbombs.Iwoulddescribemyselfashavingtakenamiddle coursebetweencompletelynaiveidealismandextremejingoism.Ifollowedmyinstincts(orperhapslackofinstincts)andwasmainlyinterestedinthescientificaspects ofthework.Theproblemsofnuclearphysicswereveryinterestingandledintonewregionsofphysicsandastrophysics.PerhapsIalsofeltthattechnologicalsequels toscientificdiscoverieswereinevitable.FinallyItrustedtheultimategoodsenseofhumanity.TheAtomicEnergyAct,asfinallyadopted,wasmuchmoresatisfactory thantheinitialproposalswhichwouldhaveleftthedevelopmentsofatomicenergyunderthesoleandcompletecontrolofthemilitary.Françoisefeltmoredubious morallyoninstinctiveandemotionalgrounds.Ialwaysfelt

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thatitwasunwiseforthescientiststoturnawayfromproblemsoftechnology.Thiscouldleaveitinthehandsofdangerousandfanaticalreactionaries.Ontheother hand,theideaofmerelymultiplyingthenumberofbombstoinfinitymadenosensewhatsoeversinceasmallfractionofthestockpilewouldbesufficienttodestroyall populationcentersontheglobeevenifitwasassumedthatamajorityofthemissilesfailedtoreachtheirtargets.IalsodidnotbelievethatRussiawouldinvade WesternEurope.Thiswasoneofthesupposedreasonsforsuperrearmament.FromtheRussianpointofview,itseemedtometherewasnopossibleadvantage. SeeingthateveninPoland,theRussianshadtroublemaintainingtheregime,IcouldnotseeanygainfortheminmakingWestGermanycommunistic.Onthecontrary, ifallofGermanywerereunitedundercommunism,itwouldhavepresentedatremendousthreattoRussia.AunitedcommunistGermanywouldinevitablyhavetriedto become"boss"ofthecommunistworld. UponourreturntoLosAlamos,weweregivenadifferentwartimeapartmentandremainedinitonlyafewmonths.JackCalkin,whowasstillthere,livedacrossthe street.TheHawkinseswerealsoquitenear,butwerepreparingtoleavesoon.Ourresumptionofamorenatural,ifmoreSpartanmodeoflifewasarefreshingchange fromtheratherartificialLosAngelesatmosphere. Asitturnedout,Ihadnotquiterecoveredallmystrengthfromthesevereillness.DuringthefirstfewweeksIwasback,Ibecametiredafterworkingonlytwoor threehoursinmyoffice.Fortunately,thisdisappearedgradually,andIbegantofeelnormalagain.Apartfromeverythingelse,thisillnesshadbeenafinancial catastrophe.Ithadleftmewithadebtofaboutfivethousanddollarsinspiteofhealthinsurance.When,inLosAngeles,itappearedthatImightdieorremain permanentlydisabledordiminished,severalofourLosAlamosfriendsandevenpersonswhoweremerelyacquaintanceslentmoneytoFrançoise.This

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touchedusverydeeply.IrepaidthemasfastasIcould.Theresttookseveralyears. Atthattime,Adam,mybrother,wasbrilliantlyconcludinghisstudiesatHarvard,andcametovisitusinLosAlamos.I,whohadbeenconditionedbytheprewar scarcityofjobs,waspessimisticabouthischancesoffindingone.WhenIaskedhimwhathisplanswere,heanswered,"I'llgetaninstructorship,ofcourse."Ifelt dubious.Hemusthavereadtheskepticismintheexpressiononmyface,becauseIsawinhiseyesthathetookmeforapessimisticolddodo.Hewasright,because heimmediatelyobtainedaninstructor'spositionatHarvardandhasremainedthereeversince.Hebecameaneminentprofessorofgovernment,andisnowdirectorof theCenterforRussianStudies.Heisalsoaprolificandsuccessfulauthorofbooksonthehistoryofcommunism.AmonghisbestknownarebiographiesofLeninand Stalin. Earlyin1948wehadtheopportunitytomovetoawingofahouseon"bathtubrow"whichhadbecomevacant.WeremainedthereuntilweleftLosAlamostwenty yearslater.ThiswasagroupofsomefiveorsixhouseswhichdatedfromtheLosAlamosRanchSchooldays.Theyweretheonlyhouseswithbathtubs—alltheother structureshadshowers.Duringthewar,theseprizedwellingshadbeenreservedforthedirectorandotherdignitaries.Fermi,Bethe,Weisskopf,andotherimportant scientistshadlivedinthemodesttemporarywartimeconstructions. ThishousewaslocateddirectlyacrossfromtheremodeledLodgewhichservedasthetown'shotelforVIPsandofficialvisitors.Webenefitedenormouslyfromthis proximity.AllourfriendsandacquaintancesvisitingLosAlamoswereonlyafewstepsaway.Itwaseasyforthemtovisitforadrink,acasualmeal,oranhourspent onourterrace.FrançoisecalledourhousetheLodgeannex.OntheirfrequentvisitstoLosAlamos,JohnnyandKlariparticularlylikedlivinginalittlecottagejustnext doortoouryard.The

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informalityofalltheseget­togetherscontributedagreatdealtothepleasantnessofourlifeinLosAlamos.Noendofscientific,politicalandpersonalconversations tookplacethere.Awholebookcouldbewrittenaboutthem. BobRichtmyerhadreplacedBetheasleaderofthetheoreticaldivision.HetooktheplaceofPlaczek,whohadheldthepostforafewmonthsafterthewar'send.I hadmetRichtmyerduringthewarwhenhevisitedLosAlamosperiodicallyfromWashingtonwhereheworkedinthepatentoffice.Hewastall,slim,intense,very friendly,andobviouslyamanofgreatgeneralintelligence.Hewasinterestedinmanyareasofmathematicsandmathematicalphysics.Laterwelearnedabouthis intensemusicalinterests,hisgreatlinguistictalents,andhisspecializedskills.Forexample,hewasverygoodincryptography.ButhewassoextremelyreservedthatI founditdifficulttoknowhimintimatelyeventhoughwewereandareonveryfriendlyterms. NorrisBradburyhadreplacedOppenheimerasdirectorofthelaboratory.Ihadmethimonlybrieflyduringthewar.Hewasapleasant,straightforward,matteroffact youngerman,eagertotakeontheresponsibilityforcontinuingthisextremelyimportantwork,eventhoughherealizedthatitwasnoteasytostepintotheshoesof RobertOppenheimerwhowasintheprocessofbecomingalegendaryfigure. Norrisdeservesallthecreditforrescuingtheprojectfromaslowdeclineintoamere"ballistics"lab.Itcouldeasilyhaveshrunkintothenarrowconfinesofaweapons arsenal,notunlikesomethatremainedintheCaliforniadesert.Underhismanagementtheintellectualandtechnologicallevelofthelaboratorybegantopickupslowly butsurely.Itbecameasolidandpermanentplacestaffedbygoodscientists,withanincreasinglybroaderrangeofinterestingscientificproblemsandthefantastic prospectsofatomic­agetechnology.(Now,underHaroldAgnew'sdirection,evenmoreso.) Norriswasratherdiffidentinhisapproachtothescientistswhohadleft.Hefeltthattheyshouldrecognizebythemselveshowimportantforthecountryandtheworld it

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wasforthemtocomeback.Asaresult,althoughhewantedto,hedidnotliketoaskpeoplelikeFermiorBetheorTellertovisit.Itwasactuallylefttome,withhis consent,towritesuchinvitations,alongwithCarsonMarkandRichtmyer.Thus,inawayIwasinstrumentalinbringingTellerbacktoLosAlamos. Thelaboratorybegantoexpandagain.Therealizationofthepoliticalimportanceofnuclearenergyforpeaceandofnuclearweaponsfordefensemadeitagainamost prominentvitalspotinnationalaffairs.Highgovernmentofficialswereagainfrequentvisitors.JimFisk,aformerJuniorFellowatHarvardandfriendofminewhohad becomeinvolvedinatomicenergyactivitiesandwashighupintheBellTelephoneResearchLabs,wasoneofthem. DuringthevonNeumanns'visits,wemadeexcursionstoSantaFeandsurroundingspots,frequentlyeatinginthesmalllocalSpanish­Americanrestaurants. OntheroadtoSantaFe,eachtimewedrovebyaplacecalledTotavi(reallymoreanamethanaplace),IwouldlaunchintoLatinandrecite,"Toto,totare,totavi, totatum,"andhewouldaddsomeformofthefuture.Thiswasoneofournonsensicalverbalgames.Anotherchildishonewastoreadroadsignsbackwards.Johnny alwaysread"pots"forstop"or''otla"for"alto"inMexico. AnothergameJohnnyandKlarilikedtoplayonthatroadwastheBlackMesagame.BlackMesawasanIndianlandmarkintheRioGrandeValleywhichwasvisible onandoffonthewaydownfromLosAlamos.Thefirstonewhospotteditcalledtheother'sattentionbyexclaiming"BlackMesa!"andscoredapoint.Thegame wentonfromjourneytojourney,pointsbeingscoredasintenniswithgamesandsets.Theyneverseemedtoforgetfromonetriptothenextwhatthescorewas. Johnnyalwayslikedthesebriefverbaldistractionsfromseriousconcentratedthought. Intheearlyyearsafterthewar,theAECstartedtobuildanelegant,permanentstructureforitsofficesandthoseofthesecurityservices,evenbeforenewandmore comfort­

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ablehousingwasreadyfortheresidents.Johnnyremarkedthatthiswasentirelyinthetraditionofallgovernmentadministrationsthroughtheages,andhedecidedto callthebuilding"ElPalaciodeSecurita."ThiswasagoodenoughmixtureofSpanish,LatinandItalian.Sotogohimonebetter,Iimmediatelynamedanewlybuilt church"SanGiovannidelleBombe." Thisisaboutthetimewemadeupa"Nebechindex."Johnnyhadtoldmetheclassicstoryofthelittleboywhocamehomefromschoolinpre­World­War­IBudapest andtoldhisfatherthathehadfailedhisfinalexamination.Thefatheraskedhim,"Why?Whathappened?"Theboyreplied,"Wehadtowriteanessay.Theteacher gaveusatheme:thepast,thepresent,andthefutureoftheAustro­HungarianEmpire."Thefatherasked,"So,whatdidyouwrite?''andtheboyanswered,"Iwrote: Nebech,nebech,nebech.""Thatiscorrect,"hisfathersaid."WhydidyoureceiveanF?""Ispellednebechwithtwobb's,"wastheanswer. Thisgavemetheideaofdefiningthenebechindexofasentenceasthenumberoftimesthewordnebechcouldbeinsertedinitandstillbeappropriate,thoughgiving adifferentflavortothemeaningofthesentenceaccordingtotheworditqualifies.Forinstance,onecouldarguethatthemostperfect"nebechthree"sentenceis Descartes'statement:Cogito,ergosum.Onecansay,Cogitonebech,ergosum.OrCogito,ergonebechsum.OrCogito,ergosumnebech.Unfortunatelythiselegant exampleoccurredtomeonlyafterJohnny'sdeath.JohnnyandIusedthisindexfrequentlyduringmathematicaltalks,physicsmeetingsorpoliticaldiscussions.We wouldnudgeoneanother,whisper"Nebechtwo"ataparticularstatement,andenjoythisgreatly. Now,ifthereaderissufficientlymystified,Iwillexplainthat"nebech"isanuntranslatableYiddishexpression,acombinationofcommiseration,scorn,drama,ridicule.

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Totrytogivetheflavoroftheword,imaginetheWilliamTellstoryasactedoutinaJewishschool.InthescenewhereWilliamTellwaitsinhidingtoshootGessler,an actorsays,inYiddish:"ThroughthisstreettheNebechmustcome."ItisobviousthatGesslerisaNebechsincehewillbethevictimofWilliamTell.Butifnebechhad beeninfrontofthewordstreet,thentheaccentwouldbeonstreet,indicatingthatitwasnotmuchofastreet.Toappreciatethismaytakeyearsofapprenticeship. SomemonthsafterIreturnedtoLosAlamos,IinvitedmyoldfriendandcollaboratorfromMadison,C.J.Everett,tojoinmeinthelaboratory.Hehadremainedin Madisonthroughthewar;Iknewfromourcorrespondencethathewasgettingtiredofteaching,soIproposedtohimtovisitandrenewourcollaboration.Hewas thefirstandonlypersonevertoarriveinLosAlamosbybusforanofficialinterview.Theprojectalwayspaidforaroometteonthetrain,orplanefare,andhis modestycausedasensation.Shortlyaftertheinterview,hemovedtoLosAlamoswithhiswifeandsonandtherebeganthecontinuationofourcollaborationin probabilitytheoryandothermathematics,thenourjointworkontheH­bomb. InMadisonhewasalreadyashyandretiringman,butastimepassedhebecamemoreandmoreofarecluse.IntheearlydaysofhisstayinLosAlamos,althoughhe wasreluctanttominglewithpeople,hecouldstillbecoaxedintocomingtoourhouseifonemadethesolemnpromisethatnooneelsewouldbethereatthesame time.Laterheevenrefusedtodothat,andnowtheonlyplaceonecanseehimisinhislittlewindowlesscubicleofanofficeorinacarrelintheexcellentlaboratory library. Oneofthelaboratoryroutineswasthepreparationofamonthlyprogressreport.Everystaffmemberhadtoturninabriefrésuméofhisworkandresearchactivity.I havealreadysaidthatEveretthadaveryexcellentsenseofhumor,

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andonemonthwhenwehadbeenexceedinglybusywithourownworkheturnedinareportwhichsaidonly,"Greatprogresswasmadeonlastmonth'sprogress report." TwoseminartalksIgaveshortlyaftermyreturnturnedouttohavegoodorluckyideasandledtosuccessfulfurtherdevelopments.Onewasonwhatwaslatercalled theMonteCarlomethod,andtheotherwasaboutsomenewpossiblemethodsofhydrodynamicalcalculations.Bothtalkslaidthegroundworkforverysubstantial activityintheapplicationsofprobabilitytheoryandinthemechanicsofcontinua. Thehydrodynamicalcalculationswereforproblemsinwhichtherewasnohopeforclosedformulaeorexplicitclassicalanalysissolutions.Theycouldbedescribedas asortof"bruteforce"calculationsusingfictitious"particles"thatwerereallynotthefluidelementsbutabstractpoints.Insteadofconsideringindividualmaterialpoints ofthefluid,itwasamatterof'usingthecoefficientsofaninfiniteseriesintowhichthecontinuumwasdevelopedasabstractparticlesforaglobaldescriptionofthe fluid.Thewholemotionisdescribedbysomeinfiniteserieswhosetermsaresuccessivelylessimportant.Consideringonlythefirstfewofthem,onechangedthepartial differentialequationsofseveralvariables(ortheintegralequationsinseveralvariables)intoordinaryortotallydifferentequationsforafinitenumberofabstract "particles.''Someyearslater,theworkofFrancisHarlowinLosAlamosdeepened,enlarged,andmultipliedthescopeofthisapproachtothecalculationsofmotions offluidsorofcompressiblegases.Thesearenowwidelyused.Thepossibilitiesofsuchmethodshavenotyetbeenexhausted;theycouldplayagreatroleinthe calculationsofairmovements,weatherprediction,astrophysicalproblems,problemsofplasmaphysics,andothers. Thesecondtalkwasonprobabilisticcalculationsforaclassofphysicalproblems.TheideaforwhatwaslatercalledtheMonteCarlomethodoccurredtomewhenI wasplayingsolitaireduringmyillness.Inoticedthatitmaybemuchmorepracticaltogetanideaoftheprobabilityofthe

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successfuloutcomeofasolitairegame(likeCanfieldorsomeotherwheretheskilloftheplayerisnotimportant)bylayingdownthecards,orexperimentingwiththe processandmerelynoticingwhatproportioncomesoutsuccessfully,ratherthantotrytocomputeallthecombinatorialpossibilitieswhichareanexponentially increasingnumbersogreatthat,exceptinveryelementarycases,thereisnowaytoestimateit.Thisisintellectuallysurprising,andifnotexactlyhumiliating,itgivesone afeelingofmodestyaboutthelimitsofrationalortraditionalthinking.Inasufficientlycomplicatedproblem,actualsamplingisbetterthananexaminationofallthe chainsofpossibilities. Itoccurredtomethenthatthiscouldbeequallytrueofallprocessesinvolvingbranchingofevents,asintheproductionandfurthermultiplicationofneutronsinsome kindofmaterialcontaininguraniumorotherfissileelements.Ateachstageoftheprocess,therearemanypossibilitiesdeterminingthefateoftheneutron.Itcanscatter atoneangle,changeitsvelocity,beabsorbed,orproducemoreneutronsbyafissionofthetargetnucleus,andsoon.Theelementaryprobabilitiesforeachofthese possibilitiesareindividuallyknown,tosomeextent,fromtheknowledgeofthecrosssections.Buttheproblemistoknowwhatasuccessionandbranchingofperhaps hundredsofthousandsormillionswilldo.Onecanwritedifferentialequationsorintegraldifferentialequationsforthe"expectedvalues,"buttosolvethemorevento getanapproximativeideaofthepropertiesofthesolution,isanentirelydifferentmatter. Theideawastotryoutthousandsofsuchpossibilitiesand,ateachstage,toselectbychance,bymeansofa"randomnumber"withsuitableprobability,thefateor kindofevent,tofollowitinaline,sotospeak,insteadofconsideringallbranches.Afterexaminingthepossiblehistoriesofonlyafewthousand,onewillhaveagood sampleandanapproximateanswertotheproblem.Alloneneededwastohavethemeansofproducingsuchsamplehistories.Itsohappenedthatcomputing machineswerecomingintoexis­

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tence,andherewassomethingsuitableformachinecalculation. Computingmachinescameaboutthroughtheconfluenceofscientificandtechnologicaldevelopments.Ononesidewastheworkinmathematicallogic,inthe foundationsofmathematics,inthedetailedstudyofformalsystems,inwhichvonNeumannplayedsuchanimportantrole;ontheotherwastherapidprogressof technologicaldiscoveriesinelectronicswhichmadeitpossibletoconstructelectroniccomputers.They,inturn,providedsuchaquantitativeincreaseinthespeedof operationsomuchgreaterthanthemechanicalrelaymachinesthatitproducedaqualitativechangeandvastlyimprovedandenlargedtheuseofthetool.Theresults arenowknowntoeveryone:computersintroducedanewageinheuristicresearch,incommunication,andinmakingthespaceagepossible. Thenumberofapplicationsinexactscience,inthenaturalsciences,andineverydaylifeissogreatthatonecantalkof"theageofcomputersandautomata"ashaving begun. Atthattimethecomputersweremerelyinstatunascendi.AsajokeIproposedtomakeMonteCarlocalculationsbyhiringseveralhundredChinesefromTaiwan, gatherthemonaboat,haveeachonesitwithanabacus,orevenjustpencilandpaper,andmakethemproducetherandomnumbersbysomeactualphysicalprocess likethrowingdice.Thensomeonewouldcollecttheresults,andtotalthestatisticsintosingleanswers. VonNeumannplayedaleadingroleinthelaunchingofelectroniccomputers.Hisuniquecombinationofgifts,hisinterests,andtraitsofcharactersuitedhimforthat role.Iamthinkingofhisability,andinclinationtogothroughallthetediousdetailsofprogramplanning,ofexecutingtheminutiaeofputtingverylargeproblemsina formtreatablebyacomputer.Itwashisfeelingforandknowledgeofthedetailsofmathematicallogicsystemsandthetheoreticalstructureofformalsystemsthat enabledhimtoconceiveofflexibleprogramming.Thiswashisgreatachievement.By

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suitableflowdiagrammingandprogramming,anenormousvarietyofproblemsbecamecalculableononemachinewithallconnectionsfixed.Beforehisinventionone hadtopulloutwiresandreconnectplugboardseachtimeaproblemwaschanged. TheMonteCarlomethodcameintoconcreteformwithitsattendantrudimentsofatheoryafterIproposedthepossibilitiesofsuchprobabilisticschemestoJohnnyin 1946duringoneofourconversations.ItwasanespeciallylongdiscussioninagovernmentcarwhileweweredrivingfromLosAlamostoLamy.Wetalked throughoutthetrip,andIremembertothisdaywhatIsaidatvariousturnsintheroadornearcertainrocks.(Imentionthisbecauseitillustrateswhatmaybemultiple storinginthememoryinthebrain,justasoneoftenrememberstheplaceonthepagewherecertainpassageshavebeenread,whetheritisontheleft­orright­hand page,upordown,andsoon.)Afterthisconversationwedevelopedtogetherthemathematicsofthemethod.ItseemstomethatthenameMonteCarlocontributed verymuchtothepopularizationofthisprocedure.ItwasnamedMonteCarlobecauseoftheelementofchance,theproductionofrandomnumberswithwhichtoplay thesuitablegames. Johnnysawatonceitsgreatscopeeventhoughinthefirsthourofourdiscussionheevincedacertainskepticism.ButwhenIbecamemorepersuasive,quoting statisticalestimatesofhowmanycomputationswereneededtoobtainroughresultswiththisorthatprobability,heagreed,eventuallybecomingquiteinventivein findingmarveloustechnicaltrickstofacilitateorspeedupthesetechniques. TheonethingaboutMonteCarloisthatitnevergivesanexactanswer;ratheritsconclusionsindicatethattheanswerissoandso,withinsuchandsuchanerror,with suchandsuchprobability—thatis,withprobabilitydifferingfromonebysuchandsuchasmallamount.Inotherwords,itprovidesanestimateofthevalueofthe numberssoughtinagivenproblem.

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Igavealotof"propaganda"talksforthismethodallovertheUnitedStates.Interestandimprovementsinthetheorycamerapidly.Hereisaneasyexampleofthis procedurewhichIoftenselected:Onemaychooseacomputationofavolumeofaregiondefinedbyanumberofequationsorinequalitiesinspacesofahighnumber ofdimensions.Insteadoftheclassicalmethodofapproximatingeverythingbyanetworkofpointsorcells,whichwouldinvolvebillionsofindividualelements,onemay merelyselectafewthousandpointsatrandomandobtainbysamplinganideaofthevalueoneseeks. Thefirstquestionsconcernedtheproductionoftherandomorpseudo­randomnumbers.Trickswerequicklydevisedtoproducetheminternallyinthemachineitself withoutrelyingonanyoutsidephysicalmechanism.(Clicksfromaradioactivesourceorfromcosmicrayswouldhavebeenverygoodbuttooslow.)Beyondthe literalor"true"imitationofaphysicalprocessonelectroniccomputers,awholetechniquebegantodeveloponhowtostudymathematicalequationswhichontheir faceseemtohavenothingtodowithprobabilityprocesses,diffusionofparticles,orchainprocesses.Thequestionwashowtochangesuchoperatorequationsor differentialequationsintoaformthatwouldallowthepossibilityofaprobabilisticinterpretation.ThisisoneofthemainthesesbehindtheMonteCarlomethod,andits possibilitiesarenotyetexhausted.IfeltthatinawayonecouldinvertastatementbyLaplace.Heassertsthatthetheoryofprobabilityisnothingbutcalculusapplied tocommonsense.MonteCarloiscommonsenseappliedtomathematicalformulationsofphysicallawsandprocesses. Muchmoregenerally,electroniccomputersweretochangethefaceoftechnology.Wediscussedthemanypossibilitiesendlessly.ButnotevenvonNeumanncould foreseetheirfulleconomicortechnologicalimpact.Theseaspectsoftheirdevelopmentwerestillintheirinfancysofarasindustrialapplicationswereconcernedwhen hedied

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in1957.Littledidweknowin1946thatcomputingwouldbecomeafifty­billion­dollarindustryannuallyby1970. AlmostimmediatelyafterthewarJohnnyandIalsobegantodiscussthepossibilitiesofusingcomputersheuristicallytotrytoobtaininsightsintoquestionsofpure mathematics.Byproducingexamplesandbyobservingthepropertiesofspecialmathematicalobjectsonecouldhopetoobtaincluesastothebehaviorofgeneral statementswhichhavebeentestedonexamples.Irememberproposingin1946acalculationofaverygreatnumberofprimitiverootsofintegerssothatbyobserving thedistributionsoneobtainedenoughstatisticalmaterialontheirappearanceandonthecombinatorialbehaviortoperhapsgetsomeideasofhowtostateandprove somepossiblegeneralregularities.Idonotthinkthatthisparticularprogramhasbeenadvancedmuchuntilnow.(Inmathematicalexploratoryworkoncomputersmy collaboratorswereespeciallyMyronSteinandRobertSchrandt.)Inthefollowingyearsinanumberofpublishedpapers,Ihavesuggested—andinsomecases solved—avarietyofproblemsinpuremathematicsbysuchexperimentingorevenmerely"observing."TheGedankenExperimente,orThoughtExperiments,of Einsteinarepossibleandoftenusefulinthepurestpartofmathematics.Oneofthepapersoutliningafieldofexplorationin"non­linearproblems"waswrittenin collaborationwithPaulStein.Bynow,awholeliteratureexistsinthisfield. Quiteearly,infactonlysomemonthsaftertheelectroniccomputercalledMANIACbecameavailableinLosAlamos,Itriedwithanumberofassociates(PaulStein, MarkWells,JamesKister,andWilliamWalden)tocodethemachinetoplaychess.Itwasnotsoterriblydifficulttocodeittoplaycorrectlyaccordingtotherules. Therealproblemisthat,eventoday,nobodyknowshowtoputinitsmemoryexperiencesofpreviousgamesandageneralrecognitionofthequalityofpatternsand positions.Neverthelessitcanbe

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madetoplaysoitcanbeatarankamateur.Werealizedthatthedifferencesbetweenplayingpoorlyandplayingwellaremuchgreaterthanteachingittomakelegal movesandrespondtoobviousthreats,andsoon.Thisgamewasplayedonasix­by­sixboardwithoutbishops(toshortenthetimebetweenmoves).Wewrotean articlethatappearedintheU.S.ChessReviewandwassoonreprintedbyaRussianchessmagazine.Stein,originallyaphysicist,"converted"tomathematicsand becameoneofmyclosestcollaborators. Curiouslythepatternsofchessremindmeoforientalrugsandalsoofsomethingthatlaymenwon'tunderstand—verycomplicatednon­measurablesets.IthinkIama fairchessplayer.WhenIfirstcametothiscountryIplayedwithothermathematiciansforrelaxation.InLosAlamosduringtheyearsafterthewar,friendsandyounger colleaguesorganizedachessclub,andIplayedmanygames.TheLosAlamoschessteamonwhichIplayedboardnumberoneseveraltimesbeatSantaFeandeven AlbuquerquewiththeirpopulationsrespectivelythreeandfifteentimesthatofLosAlamos. Itwasin1949,afterTeller'sreturn,thatGeorgeGamow,whomIhadmetbrieflyinPrincetonbeforethewar,cametoLosAlamosforalengthyvisit.Hewasona year'sleavefromGeorgeWashingtonUniversityinWashington.Inphysicalappearancehewasquiteanimpressiveman,sixfeetthreeinchestall,slimin1937(by 1949heavyset),blond,blue­eyed,youthfullooking,fullofgoodhumor.Hehadaverycharacteristicshufflingwayof'walkingwithmincingsteps.Hewasvery differentfromthepopularpictureofthespecialized,scholarlyscientist—notatallthestandardtypeofacademicpersonage.Therewasnothingdryabouthim.Atruly "three­dimensional"person,hewasexuberant,fulloflife,interestedincopiousquantitiesofgoodfood,fondofanecdotes,andinordinatelygiventopracticaljokes. Almostatoncewebecamefriendsandengagedininterminablediscussions.Insomewaysourtemperaments

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matched.Hefoundsomethingcongenialinmywayofthinking(ornotthinking)aboutproblemsofphysicsalongstandardlines.Helikedtoapproachdifferent problemsfrommanydifferentdirectionsinanunassuming,direct,andoriginalway.Hetalkedabouthimselfagreatdeal.Generallyhewasoneofthemostegocentric personsIhaveknown,yetparadoxically(becausethiscombinationissorare)hewasatthesametimecompletelydevoidofmalicetowardsothers. Itwashe(andEdwardU.Condon,independentlyandalmostsimultaneously),whostartedtheoreticalnuclearphysicsina1928paperonthequantumtheoretical explanationofradioactivity.Inscientificresearch,heconcentratedonafewgivenproblemsoveraperiodofyears,returningtothesamequestionstimeandagain. Banachoncetoldme,"Goodmathematiciansseeanalogiesbetweentheoremsortheories,theverybestonesseeanalogiesbetweenanalogies."Gamowpossessed thisabilitytoseeanalogiesbetweenmodelsforphysicaltheoriestoanalmostuncannydegree.Inourever­more­complicatedandperhapsoversophisticatedusesof mathematics,itwaswonderfultoseehowfarhecouldgousingintuitivepicturesandanalogiesfromhistoricalorevenartisticcomparisons.Anotherqualityofhiswork wasthenatureofthetopicswithwhichhedealt.Heneverallowedhisfacilitytocarryhimawayfromtheessenceofhissubjectinpursuitofunimportantdetailsand elaborations.Itwasalongthegreatlinesofthefoundationsofphysics,incosmology,andintherecentdiscoveriesinmolecularbiologythathisideasplayedan importantrole.Hispioneeringworkinexplainingtheradioactivedecayofatomswasfollowedbyhistheoryoftheexplosivebeginningoftheuniverse,the"bigbang" theory(hedislikedthetermbytheway),andthesubsequentformationofgalaxies.Therecentdiscoveryoftheradiationpervadingtheuniverse,correspondingtoa temperatureofsomethreedegreesabsolute,seemstoconfirmhispredictionin1948concerningresidualradiationfromthebigbangabouttenbillionyearsago.This discoverycameafterhisdeathin1968.

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Gamow,whowasacompletelaymaninthefieldofbiology(someofhisdetractorswouldsayalmostacharlatan),proposed,withhisfantasticallyunerringinstinct, someideasabouthowthecodereallyworked.IthinkhewasthefirsttosuggestthatthesequenceofthefoursubstancesoftheDNAdenotedbythelettersA,C,T, G,expressedwords,andhowfromthesefourlettersonecouldbuildtwentyortwenty­threeaminoacidswhich,inturn,consideredaswords,combinedintophrases definingthestructuresofproteins.Gamowhadthisideabeforeanyoneelse.Heevenalmosthadthecorrectway(laterfoundbyCrick)ofexpressingtheformationby triplets.Atfirsthethoughtfourwerenecessary.Hewasalmostcorrectfromthestart. Onemayseeinhiswork,amongotheroutstandingtraits,perhapsthelastexampleofamateurisminscienceonagrandscale. Anoverwhelmingcuriosityabouttheschemeofthingsinnature,intheverylargeandintheverysmall,directedhisworkinnuclearphysicsandincosmology. Themeaning,theoriginandperhapsthevariabilityintimeofthefundamentalphysicalconstantslikec(thevelocityoflight),h(thePlanckconstant),G(thegravitation constant)occupiedhisimaginationandhiseffortsduringthelastyearsofhislife. Thegreatunansweredquestionsconcerntherelationsbetweenmassesofelementaryparticlesandalsotheverylargenumberswhicharetheratiosbetweenthe nuclear,electrical,andgravitationalforces.Gamowthoughtthatthesenumberscouldnothavearisenasaresultofaninitialaccident,andthattheymightbeobtainable fromtopologicalornumber­theoreticalconsiderations.Hebelievedinthefinalsimplicityofatheorywhichonedaywouldexplainthesenumbers. TheFrenchfictionaldetectiveArsèneLupin,archrivalofSherlockHolmes,said:"Ilfautcommenceràraisonnerparlebonbout."(Youhavetostartthinkingfrom therightend.)Gamowhadaparticularlygreatgiftforthis.Heused

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modelsorsimiles;speakingmathematicallyhewasguidedbyisomorphismsorhomomorphisms.Abstruseideasofquantumtheoryandthemorepalpableideas betweenstructuresofclassicalphysicsweretransformedortransfigured,notjustbyrepetition,butbygoingtovariablesofhighertime,tousetechnicalmathematical termsagain. In1954GamowandIhappenedtobeinCambridge,Massachusetts,atthesametime.Iwastellinghimaboutsomeofmyspeculationsontheproblemsofevolution andthepossibilitiesofcalculatingtherateofevolutionoflife.Onedayhecametoseemeandsaid:"Let'sgotoMassachusettsGeneralHospital—thereisan interestingbiologyseminar."AndwedroveinhisMercedes.OnthewayIaskedhimwhowastalking.Hesaid,"Youare!"Apparentlyhehadtoldtheprofessors runningtheseminarthatwewouldbothtalkaboutthesespeculations.Andindeedwebothdid.OnthewayhomeIremarked,"Imagine,George,youandmetryingto talkaboutbiology!Allthesepeople,allthesedoctorsinwhitesmocks—theywerereadytoputusinstraitjackets." Inconversationsduringthelastfewmonthsofhislifeheoftenreturnedtotheconsiderationofschematathatmightpossiblythrowlightonthemysteryofthe elementaryparticlesandtheconstantsofphysics.Inadreamhehad,whichherelatedtohiswife,Barbara,shortlybeforehisdeath,hedescribedthetantalizing experienceofbeinginthecompanyofsuchgreatspiritsasNewtonandEinsteinandofdiscovering,astheyhaddiscovered,theextremesimplicityoftheultimate scientifictruths. Atthesametimethathedelightedincuttingtotheheartofthings,hekepttrackofallhismundaneactivitiesinaverydetailedandsystematicway.FromthefirsttimeI methimtotheend,whenwewerebothprofessorsonthesamecampusinBoulder,Irememberhiscollectingandputtinginorderallmannerofsnapshotsandpictures ofhisvariousactivities,markersasitwereofscientificprogress,vacationtrips,discussionswithfriends.Healsolovedtocompose

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photo­montagescombininghisowndrawingswithphotographiccut­outs.Thesewereintendedasillustrationsorcaricaturesofscientificdiscoveries. Allhiswritingsarecharacterizedbyanaturalflowofideas,asimpleuninvolvedpresentation,andaneasy,neverredundant,amusingbutneverfrivolousstyle.He wroteeasily,quickly,hardlyeverrewriting,fillinginnumerablepages,eachwithonlyafewlineshandwritteninenormouscharacters. Hisnowclassicbooksonthehistoryofphysicsandonthenewideasinthephysicalsciencesshowhimtohavebeenwithoutmaliceorharshjudgmenttowardsfellow physicists.Hewassparingwithrealpraise,reservingitonlyforthegreatachievements,buthenevercriticizedorevenpointedoutmediocrity. Hispopularbooksonsciencereceivedgreatacclaim.Amongtheoutstandingqualitiesoftheseworksaresimplicityofapproachandtheavoidanceofunnecessary technicaldetailsthatalsodistinguishedhisworkinresearch. Hishonestymadehimwriteexactlythewayhethought,embodyingthepreceptofDescartes:''orderingone'sthoughtstoanalyzethecomplexbydissectingitintoits simplerparts." OnecharacteristicofGamow,whichwasnotperhapsdirectlyvisiblebutwaseasilydeduciblefromhisconversationandhiscreativeactivity,washisexcellent memory.AfterdinneroratpartieshelovedtoreciteforthebenefitoffriendsofSlavicoriginlongexcerptsofRussianpoetry;hecouldquotePushkinorLermontov bythehour.HealsolovedtouseRussianproverbs. Gamowhadareadywitandmademanybonsmots.HetoldmethatthedayhedrovetoLosAlamosforthefirsttime,henoticedthat"asonecrossestheRio Grande,andbeforeonearrivesattheValleGrande[theValleGrandeisanextinctvolcanoofenormousdimensionsinthemountainsbehindLosAlamos],onecomes tothecityoftheBombaGrande."

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In1949myapproachingfortiethbirthdayappearedasathreateninglandmarkinmylife.Ialwaysconsidereditominoustobeslippingintomiddleage.One'sfeelings aboutagechangewithtime,ofcourse,butmathematicianshaveareputationforpeakingearly,andmany,myselfincluded,haveanadmirationforyouth.ThisHellenic accentonyouthisalsosomethingofanAmericanobsession.FrommyearliestreadingIadmiredAbel,theNorwegiangeniuswhodiedattwenty­seven,andGalois, thecreatorofnewideasinalgebraandgrouptheorywhodiedinaduelinParisattheageoftwenty­one.Thegreatestachievementsofmiddle­agedmendidnot touchmeasmuch.Thetragedy,ofcourse,isthatasonegetsolderonetendstotrytouseoldtricksinnewsituations;asortofself­poisoningstopscreativity.My friendRotasaidhedidnotbelieveitwascreativitythatstoppedbutinterest.Thatremarksparkedafeelingofdéjàvu.Iagreeinpart.Maybeitislikeboxing:itisnot thatthereactionsslowdownorthatoneismoreeasilyfatigued;whenboxersstarthavingtothinkaboutwhattheyaredoing,theylose,becausethereactionshouldbe instinctive,quickautomaticsubroutines,sotospeak. Johnnyusedtosaythataftertheageoftwenty­sixamathematicianbeginstogodownhill.WhenImethimhewasjustpastthatage.Astimewentonheextendedthe limit,butkeptitalwaysalittlebelowhisage.(Forexample,whennearforty,heraisedittothirty­five.)Thiswascharacteristicofhisratherself­effacingmanner.He didnotwanttogivetheappearanceofconsideringhimself"in."Heknewthatself­praisesoundsridiculoustoothers,andhewouldleanoverbackwardstoappear modest.I,onthecontrary,alwaystookpleasureinboasting,especiallyaboutsomeofmyowntrivialaccomplishmentslikeathleticsorwinningatgames.Children boastquitenaturally.Intheliteratureofantiquity,notablyinHomer,heroesbragopenlyabouttheirathleticprowess.Scientistssometimesboastbyimplicationwhen theycriticizeorminimizetheachievementsofothers. ApproachingfortyandthinkingaboutwhatIhadac­

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complisheduptothatpoint,Iwasstillveryhopefulthatmuchworklayaheadofme.PerhapsbecausemuchofwhatIhadworkedonorthoughtabouthadnotyet beenputinwriting,IfeltIstillhadthingsinreserve.Giventhisoptimisticnature,IfeelthiswayevennowwhenIampastsixty.

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Chapter11 The"Super" 1949–1952 IwasreturningtoLosAlamosfromoneofmyfrequenttripseastwhenourmonitoringsystemsdetectedtheRussianA­bombexplosion.Thenewshadnotyetbeen madepublic.Immediatelyuponmyarrivalseveralfriends—Metropolis,Calkin,andothers—whometmeatthelittleairport,greetedmewiththeseitemsofnews:(a) inapokergamethenightbefore,Jackhadwoneightydollars(anenormoussumforourkindofstakes),and(b)theRussianshaddetonatedanatomicbomb.I consideredthisforamomentandsaid,ofcourse,Ibelieved(a).Itwas(b)thatwastrue. JohnnywasinLosAlamosandheandTellerhadbeenspendingtimetogetherdiscussingthisominousdevelopment.IjoinedtheminJohnny'sroomattheLodge.The generalquestionwas"Whatnow?"AtonceIsaidthatworkshouldbepushedonthe"super."Tellernodded.Needlesstosay,thatwasonhismindalso.Theysaid theyhadbeendiscussinghowtogoaboutit.ThenextdayTellerleftfor

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Washington,perhapstoseeAdmiralStrauss,whowasamemberoftheAtomicEnergyCommission,todowhatpolitickinghecouldthere. StrausswasoneofthefirstAECCommissioners.HewasJewish.TalkingtohimInoticedthathehadtherathercommon—andtomepleasant—Jewishtendencyto admiresuccessfulscientists.Hepossessedasortofwistfulappreciationofscience,perhapsbecausehewasnotascientisthimself.Inhisearlydaysonthe Commissionhehadpushedforthedevelopmentofamonitoringsystemtodetectthepresenceofnuclearworkanywhereintheworld.Thiscouldbedoneby examiningairsamplesfromtheatmosphereforthepresenceofcertaingaseswhichcamefromuraniumfission.TheideacamefromTonyTurkevitch,aphysical chemistfromChicago.IrememberhismentioningsuchaplaninmypresenceinLosAlamosduringthewar. TowhatextentStrauss'scounsel,influencedbyTellerandvonNeumann,contributedtoPresidentTruman'sdecisiontoorderfull­speedworkontheH­bomb,Ido notknow. Ithastoberepeatedhereagainthattheworkonthe"super"hadbeengoingonefficientlyandsystematically.NorrisBradburywasdirectingtheallocationofthe theoreticaleffort.SomesixmonthsbeforethisnewsfromRussia,ImentionedtohimthatIhadtheimpressionthatsomepeopleinWashingtondidnotwantthiswork tocontinueandNorrishadsaid:"I'llbedamnedifI'llletanybodyinWashingtonoranypoliticianstellmewhatworknottodo."Irememberhissmilingexpression whenhesaidit.Thissentimentwasnotwhatisnowcalledhawkishormotivatedbypoliticalormilitaryconsiderations;itreferredpurelytoscientificandtechnological inquiry. Theoreticalworkonthe"super"had,asIhaveshown,continuedallalongafterthe"super"conference,butIdonotthinkTellerwantedthispublicizedverymuch,forI thinkheeitherbelievedhewasorwantedtobeknownasnotonlythemainbutthesolepromoter,defender,and

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organizerofthework.PerhapshefeltthatBradbury,asheadofthelaboratory,wouldreceivemostofthecreditforthefuturethermonuclearbomb,justas OppenheimerhadfortheA­bomb,totheexclusionoftheotherscientistswhohaddonethetechnicalwork.IndeedTellerhadbeenandwastheoriginalproponentof intensiveworkonthermonuclearexplosionsintheUnitedStates. This,ofcourse,ismyowninterpretationofthereasonsforthedevelopmentsthatfollowed.Itmaybesubstantiatedintheexistingliterature,includingthenotorious Shepley­BlairaccountpublishedinLifemagazine,whichcontributedsomuchtotheestablishmentofTellerasthe"FatheroftheH­Bomb."Theirsubsequentbook waslaterdiscreditedbecauseofthemisinformationitcontained. ShortlyafterPresidentTruman'sannouncementdirectingtheAECtoproceedwithworkontheH­Bomb,E.O.LawrenceandLuisAlvarezvisitedLosAlamosfrom BerkeleyandstarteddiscussionswithBradburyandthenwithGarnow,Teller,andmyselfaboutthe'feasibilityofconstructinga"super."Thisvisitplayedapartinthe politicsofthisenterprise. OneofTeller'sfirstmoveswastoenlistayoungphysicist,FrédéricdeHoffmann,ashisassistant.AnativeofVienna,FreddyhadcometotheUnitedStatesasa youngboybeforethewar.Hewasyoung,clever,intelligent,andquick,butnotwhatyoumightcallareallyoriginalscientist.Hebecameasortoffactotum,ajackof alltradesforcommunications,contactswithadministrators,andotherduties.HecarriedEdward'smessagesbackandforthtoWashingtonandalsodidsome technicalwork.HewasanidealassociateforEdward,wholatercouldaffordtobegenerouswithcreditforFreddy'scontributions,whichwouldnotdetractfromhis ownappearanceasthealmostexclusiveoriginator,propagator,andexecutoroftheproject. Afirstcommitteewasformedtoorganizeallworkonthe"super"andinvestigateallpossibleschemesforconstruct­

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ingit.Thecommittee'sworkwasdirectedbyTeller,aschairman,Gamow,andmyself. Severaldifferentproposalsofideasexistedonhowtoinitiatethethermonuclearreaction,usingfissionbombsasstarter.OneofGamow'swascalled"thecat'stail." AnotherwasEdward'soriginalproposal.Gamowdrewahumorouscartoonwithsymbolicrepresentationsofthesevariousschemes.Inithesqueezesacatbythetail, Ispitinaspittoon,andTellerwearsanIndianfertilitynecklace,whichaccordingtoGamowisthesymbolforthewomb,awordhepronounced"vombb."This cartoonhasappearedamongtheillustrationsinhisautobiography,MyWorldLine,publishedbyTheVikingPressin1970. BothGamowandIshowedalotofindependenceofthoughtinourmeetings,andTellerdidnotlikethisverymuch.Nottoosurprisingly,theoriginal"super"directing committeesoonceasedtoexist.AtamomentwhenbothGamowandIwereoutoftown,TellerprevaileduponBradburytodisbandthecommitteeandtoreplaceit byanotherorganizationalentity.Gamowwasquiteputoutbythis.Ididnotcare,butIwrotehim,propheticallyitseems,thatgreattroubleswouldfollowbecauseof Edward'sobstinacy,hissingle­mindedness,andhisoverwhelmingambition.Thisletter,likeallcommunicationsaboutworkinLosAlamos,was"classified."Iexpectit isstillfiledsomewhereandperhapssomedaymaybeincludedinsomecollectionofdocumentsfromthatperiod.Anothersuch"indiscreet"letterisoneIwrotetovon NeumanninwhichImadefunofEdward'sattitude.ThisletterhappenstobequotedinthesecondvolumeoftheofficialhistoryoftheAEC,TheAtomicShield.Init ImentionthatanideaoccurredtomewhichIhadcommunicatedtoEdward.IaddedinjestthatsinceEdwardlikeditverymuchperhapsthatmeantitwouldnot workeither. AtsometimeIoutlinedapossibledetailedcalculationwhichbecamethebaseoftheworkcarriedoutbyvonNeumannonthenewlybuiltelectroniccomputing machines

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withthehelpofKlari,asaprogrammer,andCerdaandFosterEvans,ahusband­and­wifeteamofphysicistswhohadjoinedtheprojectafterthewar. Thewartime,orrather,immediatepostwarMetropolis­FrankelcalculationwasveryschematiccomparedtowhatIhadinmind.Moreambitiouscalculationsofthis sorthadbecomepossiblesincecomputershadimprovedbothinspeedandinthesizeoftheirmemories.Thestepsoutlinedinvolvedafantasticnumberof'arithmetical operations.Johnnysaidtomeoneday,"Thiscomputationwillrequiremoremultiplicationsthanhaveeverbeendonebeforebyallofhumanity."ButwhenIestimated roughlythenumberofmultiplicationsperformedbyalltheworld'sschoolchildreninthelastfiftyyears,Ifoundthatthisnumberwaslargerbyaboutafactoroften! Ourswasthebiggestproblemever,vastlylargerthananyastronomicalcalculationdonetothatdateonhandcomputers,anditneededthemostadvancedelectronic equipmentavailable.BythenvonNeumann'sPrincetonMANIACwasfunctioningandaduplicateversionwasbeingbuiltinLosAlamosunderthedirectionof Metropolis. StillTellerkeptonhintingthatnotenoughworkwasbeingdoneonhisoriginalschemefortheignitionofthe"super."Hekeptinsistingoncertainspecialapproachesof hisown.ImustadmitthatIbecameirritatedbyhisinsistence;incollaborationwithmyfriendEverettonedayIdecidedtotryaschematicpilotcalculationwhichcould giveanorderofmagnitude,atleast,a"ballpark"estimateofthepromiseofhisscheme. Beforewestartedthiscalculationof'theprogressofathermonuclearreaction(burninginamassofdeuteriumordeuterium­tritiummixture),EverettandIhaddonea lotofworkonprobabilityquestionsconnectedwiththeactiveassembliesofuraniumandwithneutronmultiplications.Weworkedoutatheoryofmultiplicative processes,aswecalledit.(Nowthepreferrednameis"branchingprocesses.")Thisworkfollowedtheideasdevelopedinthe

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reportonbranchingprocesseswrittenwithDavidHawkinsduringthewar.Butitelaborated,deepened,andextendedthemverygreatly.ThereportofHawkinsand myselfconistedofafewpages.Theresultsofseveralmonths'workwithEverettwerecontainedinthreelargereportsofahundredpagesormore.Thelatterbecame thebasisformuchsubsequentwork,someofitdonelaterindependentlybyRussianandCzechscientists. Withintheformalorganizationofthelaboratory,IwastheleaderofGroupT­8,andEverettwasitsonlymember.Everyday,inhisofficeadjoiningmine,wehaddone quitealotofothermathematicalworknotnecessarilyconnectedwiththecurrentprogrammaticquestionsofLosAlamosandwewoulddiscussthe "universe"(Johnny'sexpression),mathematicalorotherwise. Nowwestartedtoworkeachdayforfourtosixhourswithsliderule,pencil,andpaper,makingfrequentquantitativeguesses,andmanagedtogetapproximate resultsmuchmorequicklythanthegiganticproblem,whichwasprogressingslowly.Muchofourworkwasdonebyguessingvaluesofgeometricalfactors,imagining intersectionsofsolids,estimatingvolumes,andestimatingchancesofpointsescaping.Wedidthisrepeatedlyforhours,liberallysprinklingtheguesseswithconstant slide­rulecalculations.Itwaslongandarduouswork,theresultsofwhichgavearatherdiscouragingpictureofthefeasibilityoftheoriginal"super"scheme.Our calculationshoweditsenormouspracticaldifficultiesandthrewgravedoubtsontheprospectsofEdward'soriginalapproachtotheinitialignitionconditionsofthe "super." Weproceededsomethinglikethis:eachmorningIwouldattempttosupplyseveralguessesastothevalueofcertaincoefficientsreferringtopurelygeometrical propertiesofthemovingassemblyinvolvingthefateoftheneutronsandotherparticlesgoingthroughitandcausing,inturn,morereactions.Theseestimateswere interspersedwithstepwisecalculationsofthebehavioroftheactualmo­

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tions.Thereadershouldrealizethattherealtimesfortheindividualcomputationalstepswereshort,eachlessthana"shake,"andthespatialsubdivisionsofthe materialassemblyverysmallinlineardimensions.Eachsteptookafractionofa"shake."A"shake"wasthenamegiveninLosAlamosduringthewartothetime intervalof10­8seconds.Anotherunitwasthatofacrosssectioncalleda"barn'';itwas10­24ofasquarecentimeter,aterriblysmallarea.Thenumberofindividual computationalstepswasthereforeverylarge.Wefilledpageuponpageofcalculations,muchofitdonebyEverett.Intheprocesshealmostworeouthisownslide rule,andwhenourresultswereachieved,afterseveralmonthsofwork,Everettjokedthat"thegratefulgovernmentcouldatleastoffertobuyhimanewsliderule."I donotknowhowmanyman­hourswerespentonthisproblem. Towritethereportsweenlistedthehelpofprofessionalcomputers,JosephineElliottamongthem.EvenFrançoisewaspressedintoservice,grindingoutuntold numbersofarithmeticaloperationsondeskcalculators. Lengthyasthisprocesswas,theworkwasfinishedseveralmonthsbeforethePrincetonelectroniccomputer'sresultsstartedcomingin.Thisso­to­speakhomespun partoftheH­bombdevelopmentwasdescribedinmanyofficialandpopularaccounts.Itcaughtthepubliceyeprobablybecauseofthecertainappealofits"man versusmachine"element. AsourcalculationprogresseditnaturallyattractedquitealotofattentionamongthephysicistsTellerwastryingtointerestinthe"super"project,andalsoamongthose Bradburyhadalreadyenlistedforthiswork.Distinguishedvisitorswouldappearperiodicallytoseehowthecalculationsweregoing.JohnWheeler'sfirstvisitstoLos Alamosdatefromaboutthisperiod. OnedayFermiandRabicametoouroffice,andweshowedthemtheresultswhichpointedtothemediocreprogressofthereaction.Theseresultscouldonlybe indica­

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tiveandwerebynomeanscertainbecauseofthecrudeapproximationsandguesseswhichweusedintheplaceofvoluminousnumericaloperations. WhenitappearedthatthetechnicaldifficultiesofTeller'soriginalideascouldjustifysomeofthescientificandpoliticalobjectionsofcertainphysicists,andeven perhapsthereluctanceoftheGeneralAdvisoryCommittee,HansBetheevidencedarenewedinterestinthewholeprojectandcametovisitLosAlamosmoreoften. Withhiswonderfulvirtuosityinmathematicalphysicsandwithhisabilitytosolveanalyticalproblemsofnuclearphysicshehelpedsignificantly.Afterall,itwasBethe whofirstsuggested(WeizsäckerinGermanyhadreachedthesameconclusionindependently)thatnuclearreactionsintheinteriorofthesuncouldberesponsiblefor thesun'senergygenerationandthusexplaintheradiationemittedbythesunandbyotherstars.Theiroriginal"carbonreactionmechanism"hasbeenfoundnottobe quitesoexclusivelyresponsiblefortheenergygenerationinallstarsaswasoriginallythought. Tellerwasnoteasilyreconciledtoourresults.Ilearnedthatthebadnewsdrovehimoncetotearsoffrustration,andhesufferedgreatdisappointment.Ineversaw himpersonallyinthatcondition,buthecertainlyappearedgluminthosedays,andsowereotherenthusiastsoftheH­bombproject.Subduedanddepressed,he wouldvisitourofficesperiodically,andwouldattempttoproveuswrongbytryingtofindmistakes.Oncehesaid,"Thereisamistakeherebyafactor104."This especiallyannoyedEverettwhodidnothavemuchself­confidenceasaphysicistbutwho,asamathematician,amazinglyenoughnevermademistakes.Heusedto say,"Inevermakemistakes"andthiswastrueinthatheneverusedawrongsignormadesimplenumericalmistakes,asmathematiciansoftendo.Buteachtimehe tried,Edwardhadtoadmitthatitwashewhowasatfaultinhisarithmetic. AstheresultsofthevonNeumann­EvanscalculationonthebigelectronicPrincetonmachineslowlystartedtocome

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in,theyconfirmedbroadlywhatwehadshown.There,inthecourseofthecalculation,inspiteofaninitial,hopefullooking"flareup,"thewholeassemblystarted slowlytocooldown.EveryfewdaysJohnnywouldcallinsomeresults."Iciclesareforming,"hewouldsaydejectedly. Thesecomputationswerethebestonecoulddotheoreticallyinthosedays.Becausetheexistingexperimentalvaluesfortheconstantswhichhadtobeappliedinthe calculationsforthecrosssectionswereuncertain,theprojectwasstillalive,butitwasnecessarytosearchforalternativeapproachestoignition. AllalongJohnnywasemotionallyinvolvedinfavoroftheconstructionofanH­bomb.Hehopedthatinonewayoranotheragoodschemewouldbefound,andhe neverlostheartevenwhenthemathematicalresultsfortheoriginalapproachwerenegative. Duringthiscrucialperiodofuncertainty,IvisitedhiminPrinceton.Fermihappenedtobethere,too,onabriefvisit,andwediscussedtheprospectsallafternoon, duringdinnerinJohnny'shouse,andallevening.ThenextdaywetalkedwithOppenheimer.HeknewabouttheresultsobtainedbyEverettandme.Heseemedrather gladtolearnofthedifficulties,whereasvonNeumannwasstillsearchingforwaystorescuethewholething.Johnnyoutlinedsomehydrodynamicalcalculations.Fermi concurred.Theycametoestimateacertainvelocityofexpansionwhichseemedtomemuchtooslow.WiththeexperienceofalltheworkIhaddoneinthepast months,Inoticedthattheyhaderredinassumingthedensityofliquiddeuteriumtobe1,whereasitisonlyasmallfractionof1.Thiserrorofperunitmassinsteadof perunitvolumemadethevelocityappearindeedsmallerandJohnnyrealizeditandexclaimed:"Ohmy!Itisindeedmuchfasterthanatrain."Oppenheimerwinkedat me.HelikedhavingthedifficultiesconfirmedandenjoyedcatchingvonNeumannandFermiinasmallandtrivialarithmeticerror. MycalculationswithEverettconcernedthefirstphaseof

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theexplosion,theproblemoftheinitialignition.Animportantpartofthestoryhasbeenoverlookedintheofficialaccountsandconcernssomequitefundamentalwork thatFermiandIdidfollowingthefirstcalculationoftheprogressofthereaction,itspropagationandexplosion.Innumerousjointdiscussionsweoutlinedthe possibilitiesofpropagation,assumingthatsomewayorother(perhapsbytheexpenditureoflargeamountsoftritium)theinitialignitioncouldbeachieved.Thereagain wehadtouseguessesinplaceoftheenormouslydifficultdetailedcalculationsthatwouldhaverequiredcomputersevenfasterthanthoseinexistence.Wedidthis againintime­stepstageswithintuitiveestimatesandmarveloussimplificationsintroducedbyFermi. Thenumericalworkwasdoneondeskcomputerswiththeassistanceofanumberofprogrammersfromthelaboratory'scomputinggroup,managedbyagood­ humoredNewYorker,MaxGoldstein.MuchtoMax'sannoyance,Fermiwantedtoencouragethegirlstousesliderules;themachineprecisionwasnotreally warrantedbecauseofoursimplifications.ButMaxinsistedontheusualroutineswithdeskcalculators.ReadingfromsliderulesandusinglogarithmsasFermididwas muchlessaccurate,butwithhismarveloussensehehadtheabilitytojudgetherightamountof'accuracywhichwouldbemeaningful.Thegirls,whoweremerely makingcalculationswithoutknowingthephysicsorgeneralmathematicsbehindthem,couldnotdothat,ofcourse,soinawayMaxwasquiterightininsistingonthe standardroutines. Iparticularlyrememberoneoftheprogrammerswhowasreallybeautifulandwellendowed.Shewouldcometomyofficewiththeresultsofthedailycomputation. Largesheetsofpaperwerefilledwithnumbers.Shewouldunfoldtheminfrontofherlow­cutSpanishblouseandask,"Howdotheylook?"andIwouldexclaim, "Theylookmarvelous!"totheentertainmentofFermiandothersintheofficeatthetime.

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AjointreportwaswrittenbyFermiandmyself.Enricoexercisedverygreatcautioninitsconclusions.Infact,oneconclusionstatingtheunpromisingnatureofthe reactionasitwasplannedcontainedthissentence:"Ifthecrosssectionsforthenuclearreactionscouldsomehowbetwoorthreetimeslargerthanwhatwasmeasured andassumed,thereactioncouldbehavemoresuccessfully." IbelievethisworkwithFermitohavebeenevenmoreimportantthanthecalculationsmadewithEverett.Itturnedouttobebasictothetechnologyofthermonuclear explosions.Fermiwassatisfiedwithbothitsexecutionandwiththefactthatitputalimittothesizeofsuchexplosions.Ashesaid:"Onecannotmaketreesgrow skywardindefinitely." InthemeantimeTellercontinuedtobeveryactivebothpoliticallyandorganizationallyatthemomentwhenthingslookedattheirworstforhisoriginalwartime"super" design,evenwiththemodificationsandimprovementsheandhiscollaboratorshadoutlinedintheinterveningperiod. PerhapsthechangecamewithaproposalIcontributed.Ithoughtofawaytomodifythewholeapproachbyinjectingarepetitionofcertainarrangements. Unfortunately,theideaorsetofideasinvolvedisstillclassifiedandcannotbedescribedhere. PsychologicallyitwasperhapsprecipitatedbyamemorandumfromDarolFroman,anassociatedirectorofthelaboratory,whoaskedvariouspeoplewhatshouldbe donewiththewhole"super"program.WhileexpressingdoubtsaboutthevalidityofTeller'sinsistenceonhisownparticularscheme,IwrotetoFromanthatone shouldcontinueatallcoststhetheoreticalwork,thatawayhadtobefoundtoextractgreatamountsofenergyfromthermonuclearreactions. ShortlyafterrespondingIthoughtofaniterativescheme.AfterIputmythoughtsinorderandmadeasemi­concretesketch,IwenttoCarsonMarktodiscussit. Mark,whowasbythenheadofthetheoreticaldivision,wasalreadyin

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chargeoftheveryextensivetheoreticalworksupportingTeller'sandWheeler'sspecialgroups.ThesameafternoonIwenttoseeNorrisBradburyandmentionedthis scheme.Hequicklygraspeditspossibilitiesandatonceshowedgreatinterestinpursuingit.Thenextmorning,IspoketoTeller.Idon'tthinkhehadanyreal animositytowardmeforthenegativeresultsoftheworkwithEverettsodamagingtohisplans,butourrelationshipseemeddefinitelystrained.AtonceEdwardtook upmysuggestions,hesitantlyatfirst,butenthusiasticallyafterafewhours.Hehadseennotonlythenovelelements,buthadfoundaparallelversion,analternativeto whatIhadsaid,perhapsmoreconvenientandgeneralized.Fromthenonpessimismgavewaytohope.InthefollowingdaysIsawEdwardseveraltimes.We discussedtheproblemforabouthalfanhoureachtime.Iwroteafirstsketchoftheproposal.Tellermadesomechangesandadditions,andwewroteajointreport quickly.Itcontainedthefirstengineeringsketchesofthenewpossibilitiesof'startingthermonuclearexplosions.Wewroteabouttwoparallelschemesbasedonthese principles.ThereportbecamethefundamentalbasisforthedesignofthefirstsuccessfulthermonuclearreactionsandthetestinthePacificcalled"Mike."Aflurryof activityensued.Tellerlostnotimeinpresentingtheseideas,perhapswithmostoftheemphasisonthesecondhalfofourpaper,ataGeneralAdvisoryCommittee meetinginPrincetonwhichwastobecomequitefamousbecauseitmarkedtheturningpointinthedevelopmentoftheH­bomb.Amoredetailedfollow­upreportwas writtenbyTelleranddeHoffmann.NewphysicistswerebroughttoLosAlamos,andworktowardexperimentalverificationstartedinearnest. JohnWheelercametoNewMexicotohelpTeller.Hebroughtwithhimseveralofhisbrighteststudents.AmongthemwasKenFord,withwhomIwastodo unrelatedworklateron.TherewasJohnToll,nowPresidentoftheStateUniversityofNewYorkatStonyBrook,thenapromisingyoungphysicist;Marshall Rosenbluth,whohadbeeninLos

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AlamosduringthewarasasoldierwiththeSED;TedTaylor,whocontributedsomanynewideastoatomicfissionbombs;abrilliantmathematicalphysicist,Conrad Longmire;andothertalentedyoungpeople.Intense,fastworkwenton,andtheplansfor"Mike"werereadyjustafewmonthsafterourfatefulconversation. DuringtheirLosAlamosyear,theWheelerslivedinahousenexttooursandwesawthemfrequently.Wheelerwasaveryinterestingtypeofphysicist.Tomymind, hehadalltherightdesiresfornoveltyintheoreticalideaswithoutstickingtoorigidlytopreconceivednotionsandexistingschemes.Sometimeshethoughtof outlandish­soundingschemesinphysicsorincosmology,somuchsothatsomeofhisideaswouldstrikemeaslackinginanelementofcommonsenseorconnection withpossibleexperiments.Orperhapstheywere,onthecontrary,asPaulioncesaidaboutsomeideasproposedbyHeisenberg,"notcrazyenough."Wheeler'sgreat meritishisworkongeneralrelativity,pursuedtoextremesituationslikeblackholesandbeyond;healsohasgreatdidacticaltalents.Ofhisstudents,thebestbyfar,I think,isFeynman.Longbefore,theywroteaverynicejointpaperonageneralizationofMach'sprinciple. Despitethegreatflurryofexcellentexperimentsandthethermonuclearexplosionitself,Tellercontinuedtobedissatisfiedandengagedinmultipleactivitiesinaneffort toputstillmoreoftheworkunderhiscontrol.HeexpressedgreatunhappinesswiththewayLosAlamoshandledthedevelopments,thoughBradburyandother seniormembersofthelaboratorycouldseenootherrationalwayofdoingthings.TheriftgrewsolargethatTellerputonallthepoliticalpressurehecouldmusterto startarivallaboratory.ThankslargelytohisinfluencewithLewisStraussandtheCommissioninWashington,heobtainedfundsandauthorizationtostartandstaff anotherlaboratoryinLivermore,California,ataboutthetimeoftheverysuccessful"Mike"testwhichmorethanconfirmedthepossibilities.SoLosAlamoswenton tobuildthefirstH­bombwithouthim,whilesome

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ofthefirstdesignsemanatingfromLivermorewerequiteunsuccessful.Johnnywasawareofthefeelingbetweenthetwolaboratories.Afterthefirstunsuccessful LivermoretryatathermonuclearexplosioninthePacificprovinggroundshelaughedandsaidtome:"TherewillbedancinginthestreetsofLosAlamostonight." Contrarytothosepeoplewhowereviolentlyagainstthebombonpolitical,moralorsociologicalgrounds,Ineverhadanyquestionsaboutdoingpurelytheoretical work.Ididnotfeelitwasimmoraltotrytocalculatephysicalphenomena.Whetheritwasworthwhilestrategicallywasanentirelydifferentaspectoftheproblem—in factthecruxofahistorical,politicalorsociologicalquestionofthegravestkind—andhadlittletodowiththephysicalortechnologicalproblemitself.Eventhe simplestcalculationinthepurestmathematicscanhaveterribleconsequences.Withouttheinventionoftheinfinitesimalcalculusmostofourtechnologywouldhave beenimpossible.Shouldwesaythereforethatcalculusisbad? Ifeltthatoneshouldnotinitiateprojectsleadingtopossiblyhorribleends.Butoncesuchpossibilitiesexist,isitnotbettertoexaminewhetherornottheyarereal?An evengreaterconceitistoassumethatifyouyourselfwon'tworkonit,itcan'tbedoneatall.Isincerelyfeltitwassafertokeepthesemattersinthehandsofscientists andpeoplewhoareaccustomedtoobjectivejudgmentsratherthaninthoseofdemagoguesorjingoists,orevenwell­meaningbuttechnicallyuninformedpoliticians. AndwhenIreflectedontheendresults,theydidnotseemsoqualitativelydifferentfromthosepossiblewithexistingfissionbombs.AfterthewaritwasclearthatA­ bombsofenormoussizecouldbemade.Thethermonuclearschemeswereneitherveryoriginalnorexceptional.SoonerorlatertheRussiansorotherswould investigateandbuildthem.Thepoliticalimplicationswereuncleardespitethehullabalooandexaggerationsonbothsides.Thatsinglebombswereabletodestroythe largestcitiescouldrenderall­outwarslessprobablethan

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theywerewiththealreadyexistingA­bombsandtheirhorribledestructivepower. AftercompletingthistheoreticalworkIconsideredmyjobdoneanddecidedtochangesurroundingsforawhile.Iacceptedaninvitationtospendasemesterat Harvardasavisitingprofessor.Itwasthesummerof1951.TheFermislivedintheotherhalfofourduplex.Wesawthemoften.InSeptember,asIwaspreparingto leave,Iwaspackingandworkingoncorrespondence,booksandpapersandforgottoattendanimportanteveningmeetinginBradbury'sofficeontheplanningof futureworkandexperiments.ThenextmorningIlearnedthattherehadbeenaseriesofheatedexchangesbetweenTellerandBradbury,andthatsomeacrimonious remarksbyotherscientistspresenthadbeendirectedatTeller'sratherwildaccusations.AsIcommentedonthistoFermi,herepliedwithhisusualserene imperturbability:"Whyshouldyoucare,youaregoingawaythedayaftertomorrow."Someofmyfriendsweregreatlyimpressedbythisdisplayofolympian detachment.RabiinparticularadmiredEnrico'slogicallycalmattitudes. TheOppenheimerAffair,whichgrewoutoftheviolenthydrogen­bombdebate­eventhoughtheanimositybetweenStraussandOppenheimerhadpersonaland perhapspettyorigins—didgreatlyaffectthepsychologicalandemotionalroleofscientists. OnceIaskedJohnnywhetherhethoughtthatEinsteinwouldhaveactivelydefendedOppenheimerduringthelatter'stroubles.Johnnyrepliedthathebelievednot;he thoughtEinsteinhadhadgenuinelymixedfeelingsaboutsomeofOppenheimer'sactionsandabouttheAffair. Itishardtoguessanother'smotives.Theymaybetheresultoflong­heldconvictions,politicalorientation,orevenpetscientificorphilosophicalideas.Ibelieve,for example,thatperhapssomeofthereasonsforOppenheimer'soppositiontothedevelopmentoftheH­bombwerenotexclusivelyonmoral,philosophical,or humanitariangrounds.Imightsaycynicallythathestruckmeassomeonewho,having

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beeninstrumentalinstartingarevolution(andtheadventofnuclearenergydoesmeritthisappellation),doesnotcontemplatewithpleasurestillbiggerrevolutionsto come. AnatoleFrancetellssomewherethatonedayinaparkinParishesawanoldmansittingonabenchreadinganewspaper.Suddenlyagroupofyoungstudents appeared,marchinginparadeformationandshoutingrevolutionaryslogans.Theoldmanbecameveryagitated,shakinghiscaneandshouting:"Order!Police!Police! Stop!"Francerecognizedtheoldman;inthepasthehadbeenafamousrevolutionary. Oppenheimerhadmanyunusuallystrong,interestingqualities;butinsomewayhewasaverysadman.Thetheoreticaldiscussionwhichheproposedoftheso­called neutronstarsisoneofhisgreatcontributionstotheoreticalphysics,butitsverificationwiththediscoveriesofpulsarstars,whicharefast­rotatingneutronstars,came yearsafterhisdeath. ItseemstomethiswasthetragedyofOppenheimer.Hewasmoreintelligent,receptive,andbrilliantlycriticalthandeeplyoriginal.Alsohewascaughtinhisownweb, awebnotofpoliticsbutofphrasing.Perhapsheexaggeratedhisrolewhenhesawhimselfas"PrinceofDarkness,thedestroyerofUniverses."Johnnyusedtosay, "Somepeopleprofessguilttoclaimcreditforthesin." Manyaccountsoftheseeventshavebeenwritten.Someareexaggeratedordistorted;others,liketheofficialhistoryoftheAEC,areratherobjective.Butnonecan becompleteyet,andofcoursetheeventsasseenbytheparticipantsappearindifferentlights.ThisismyownaccountofthehistoryoftheH­bombasIliveditandto theextentthatIwasdirectlyinvolvedinit.

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Chapter12 TheDeathofTwoPioneers 1952–1957 Afterthesomewhatfranticworkonthe''super"firstwithEverettthenwithFermi,andmyreturnfromthesemester'sleaveofabsenceatHarvardrenewingcontacts witholdmathematicalfriends,mypre­occupationsturnedtootherandmorepurelyscientificproblems. Computerswerebrand­new;infacttheLosAlamosMANIACwasbarelyfinished.ThePrincetonvonNeumannmachinehadmetwithtechnicalandengineering difficultiesthathadprolongeditsperfection.TheLosAlamosmodelhadbeenluckier,foritwasinthecapablehandsofJamesRichardson,anengineerinthe Metropolisgroup. Assoonasthemachineswerefinished,Fermi,withhisgreatcommonsenseandintuition,recognizedimmediatelytheirimportanceforthestudyofproblemsin theoretical

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physics,astrophysics,andclassicalphysics.Wediscussedthisatlengthanddecidedtoattempttoformulateaproblemsimpletostate,butsuchthatasolutionwould requirealengthycomputationwhichcouldnotbedonewithpencilandpaperorwiththeexistingmechanicalcomputers.Afterdeliberatingaboutpossibleproblems, wefoundatypicalonerequiringlong­rangepredictionandlong­timebehaviorofadynamicalsystem.Itwastheconsiderationofanelasticstringwithtwofixedends, subjectnotonlytotheusualelasticforceofstrainproportionaltostrain,buthaving,inaddition,aphysicallycorrectsmallnon­linearterm.Thequestionwastofindout howthisnon­linearityafterverymanyperiodsofvibrationswouldgraduallyalterthewell­knownperiodicbehaviorofbackandforthoscillationinonemode;how othermodesofthestringwouldbecomemoreimportant;andhow,wethought,theentiremotionwouldultimatelythermalize,imitatingperhapsthebehavioroffluids whichareinitiallylaminarandbecomemoreandmoreturbulentandconverttheirmacroscopicmotionintoheat. JohnPasta,arecentlyarrivedphysicist,assistedusinthetaskofflowdiagramming,programming,andrunningtheproblemontheMANIAC.Fermihaddecidedto trytolearnhowtocodethemachinebyhimself.Inthosedaysitwasmoredifficultthannow,whentherearesetrules,ready­madeprograms,andtheprocedureitself isautomated.Onehadtolearnmanylittletricksinthoseearlydays.Fermilearnedthemveryquicklyandtaughtmesome,eventhoughIalreadyknewenoughtobe abletoestimatewhatkindofproblemscouldbedone,theirdurationinnumberofsteps,andtheprinciplesofhowtheyshouldbeexecuted. Ourproblemturnedouttohavebeenfelicitouslychosen.TheresultswereentirelydifferentqualitativelyfromwhatevenFermi,withhisgreatknowledgeofwave motions,hadexpected.Theoriginalobjectivehadbeentoseeatwhatratetheenergyofthestring,initiallyputintoasinglesinewave(thenotewasstruckasone tone),wouldgraduallydevelophighertoneswiththeharmonics,andhowtheshape

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wouldfinallybecome"amess"bothintheformofthestringandinthewaytheenergywasdistributedamonghigherandhighermodes.Nothingofthesorthappened. Tooursurprisethestringstartedplayingagameofmusicalchairs,onlybetweenseverallownotes,andperhapsevenmoreamazingly,afterwhatwouldhavebeen severalhundredordinaryupanddownvibrations,itcamebackalmostexactlytoitsoriginalsinusoidalshape. IknowthatFermiconsideredthistobe,ashesaid,"aminordiscovery."AndwhenhewasinvitedayearlatertogivetheGibbsLecture(agreathonoraryeventatthe annualAmericanMathematicalSocietymeeting),heintendedtotalkaboutthis.Hebecameillbeforethemeeting,andhislecturenevertookplace.Buttheaccountof thiswork,withFermi,Pastaandmyselfasauthors,waspublishedasaLosAlamosreport. Ishouldexplainthatthemotionofacontinuousmediumlikeastringisstudiedonamachinebyimaginingthestringtobecomposedofafinitenumberofparticles—in ourcase,sixty­fouroronehundredtwenty­eight.(Itisbettertotakeapoweroftwoforthenumberofelements,whichismoreconvenienttohandleonthe computer.)Theseparticlesareconnectedtoeachotherbyforceswhicharenotonlylinearintermsoftheirdistancebutbyadditionalsmallnon­linearquadraticterms. Thenthemachinequicklycomputesinshorttime­stepsthemotionofeachofthesepoints.Afterhavingcomputedthis,itgoestothenexttime­step,computesthenew positions,andsoonformanytimes.Thereisabsolutelynowaytoperformthisnumericalworkwithpencilandpaper;itwouldliterallytakethousandsofyears.An analyticclosed­formsolutionusingthemathematicaltechniquesofclassicalanalysisofthenineteenthandtwentiethcenturiesisalsocompletelyunlikely. Theresultsweretrulyamazing.Thereweremanyattemptstofindthereasonsforthisperiodicandregularbehavior,whichwastobethestartingpointofwhatisnow alargeliteratureonnon­linearvibrations.MartinKruskal,a

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physicistinPrinceton,andNormanZabuski,amathematicianatBellLabs,wrotepapersaboutit.Later,PeterLaxcontributedsignallytothetheory.Theymade interestingmathematicalanalysesofproblemsofthissort.Amathematicianwillknowthattheso­calledPoincaréreturn­typeofdynamicalsystemcontainingthatmany particlesisterrificallylong—onanastronomicalscale—andthefactthatitcamebacksosoontoitsoriginalformiswhatissosurprising. AnotherLosAlamosphysicist,JimTuck,wascurioustoseeifafterthisnearreturntotheoriginalposition,anotherperiodstartedagainfromthisconditionandwhatit wouldbeafterasecond"period."WithPastaandMetropolis,hetrieditagainand,surprisingly,thethingcameback,apercentorsolessexactly.Thesecontinued and,aftersixortwelvesuchperiods,itstartedimprovingagainandasortofsuperperiodappeared.Againthisismostpeculiar. Otherauthors,amongthemseveralRussianmathematicians,havestudiedthisproblemandwrittenpapersaboutit.LastyearIreceivedarequestfromtheJapanese AcademyforpermissiontoreprinttheFermi­Pasta­Ulampaper.Iassentedwithouthesitationandshortlythereafterawholevolumeappearedcontainingstudiesof thesequestionsbymanyauthors. ImightsayherethatJohnPastawasaveryinterestingperson.Aphysicistbyprofession,hespentseveralyearsduringthedepressionasapolicemanonabeatin NewYorkCity.HejoinedmygroupinLosAlamos.Onthewholeverytaciturn,hecouldoccasionallymakeverycaustic,humorousremarks.WhenJohnnybecame anAECCommissioner,impressedbyPasta'scommonsense,ability,andknowledgeoftheLosAlamosscene,heinvitedhimtojointheAECinWashington. AsforJamesTuck,hewasaBritishphysicistwhohadcometoLosAlamoswiththeBritishMissionduringthewar.HehadreturnedtoOxfordafterthewar,butthen camebacktojointhelaboratoryagain.Wecollaboratedona

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methodforobtainingenergyfromfusioninanon­explosivewayandduringthewarhadwrittenajointreportonthiswhichmaystillbeclassified. AsaveryyoungphysicistTuckwasforatimeassistanttoLindemann,wholaterbecameLordCherwell,Churchill'sscienceadvisor.Hehasafundofinterestingand amusingstoriesaboutthisexperience,andhestilldefendsCherwellvigorouslyagainstaccusationsorcriticisms.HeremindsmeoftheEnglisheccentricsdescribedby JulesVerneandbyKarlMay.Verytall,movinginanabrupt,somewhatuncoordinatedway,byhisawkwardnesshecausesmanyamusingincidentsthatarealways thedelightofhisfriends.FormanyyearsTuckdirectedaLosAlamosprogramforthepeacefulusesoffusion.Thelaboratoryisstillvigorouslyengagedinalarge efforttofindmethodstoextractenergy"peacefully"fromthefusionofdeuterium. TherewasanotherproblemwhichFermiwantedtostudybutwhichwesomehownevercametoformulatewellortoworkon.Hesaidoneday,"Itwouldbe interestingtodosomethingpurelykinematical.Imagineachainconsistingofverymanylinks,rigid,butfreetorotatearoundeachother.Itwouldbecurioustosee whatshapesthechainwouldassumewhenitwasthrownonatable,bystudyingpurelytheeffectsoftheinitialenergyandtheconstraints,noforces." DuringtheseyearsweandthevonNeumannsstartedthepracticeofspendingChristmastogether.Claire,ourdaughter,wasasmallchild,anditbecameatradition thatonChristmasEveJohnnyandKlariwouldhelpusassemblehertoys.Irememberalargecardboarddollhousewhichtookmanyhours;bothJohnnyandI, especiallyI,beingineptatfollowingtheinstructionswhichcalledforinsertingtabAintoslotB.TothisdayIamincapableoffollowingwritteninstructions,whetherfor fillingoutformsorassemblingparts.Johnny,ontheotherhand,lovedit.InPrincetonheactivelyfollowedthesmallestdetailsoftheconstructionofthePrinceton MANIAC.AccordingtoBigelow,itsengi­

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neer,Johnnyhadlearnedalltheelectronicpartsandsupervisedtheirassembly.Whenthemachinenearedcompletion,Irememberhowoncehemadefunofitathis ownexpense.Hetoldme,"Idon'tknowhowreallyusefulthiswillbe.ButatanyrateitwillbepossibletogetalotofcreditinTibetbycoding'OmManePadme Hum'[Oh,thoufloweroflotus]ahundredmilliontimesinanhour.Itwillfarexceedanythingprayerwheelscando." AnotherChristmaswespenttogetherwasthatof1950.Tocelebratetheendofthedecadeandthefirsthalfofthecentury,Françoise,Claire,andItookashort vacationwiththevonNeumannsatGuayamas,Mexico.TheydroveallthewayfromPrinceton,andwearrangedtomeetinLasCrucesinthesouthernpartofNew Mexicotocontinuethetriptogether.LasCruceshadanold1890brothelwhichhadbeenremodeledintoahotelafterthewar,andweallstayedthere.Therooms werefurnishedinperiodstyleandoneachdoor,insteadofaroomnumber,thenameofagirlwasinscribed:Juanita,Rosalia,Maria.Inthemiddleofthelobby,a swingwassuspendedfromtheceiling.Theladiesapparentlyclimbedonitfromtheinteriorbalcony.ToJohnnyandmeitlookedlikethewell­knownFoucault pendulum,andweindulgedinalearnedandimproperbilingualjokewhichIwillrefrainfromrepeating. OnthedrivetoGuayamaswealsoamusedourselveswiththedevelopingofalanguagewhichwecalled"neo­castillian."InourignoranceofSpanish,itconsistedof EnglishwordswithLatinendings,forexampleelglaso,forglass.Toourgreatsurpriseandfunwefoundoutthatitworkedforsomewords.Terry'sguidetoMexico, particularlyitsprose,provideduswithmanyhoursoffuntoo.Therewasanespeciallyeloquentpageabouttheenchanted"paradisebosque"ofSonora,whichon drivingacrosswefoundtobeamiserablegroveoftreesonadrysandyterrain,unoccupiedbyany''numerousanddiverseexotictropicalbirds."Thisbecamea proverbialexpressiondenotingdisappointment.Wheneverweheardsomethingwhichdidnotquitemeasureuptoex­

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pectationseitherinmathematicsorphysics,weexchangedknowinglythewords"bosqueencantado." LongbeforeSputnik,around1951or1952IattendedanearlyICBMrocketrymeetinginWashington.Altogethertheremusthavebeentwentyormorepeople present.Gamowwasoneoftheimportantparticipants.JohnnyandTellerwerethere,too.ItwasaclassifiedmeetinginoneoftheroomsinthePentagon.Johnnywas sittingnexttomeatalongtable.Oneproblemunderdiscussionwasonhowtoguiderockets.Tellersuggestedachemicalpathtoatarget.Gamowcalledit"smelling" theway.Otherpeoplesuggestedotherschemes.Iproposed"ballistic"projectiles,whosetrajectorycouldbecorrectedifneedbeseveraltimesalongtheway.I rememberJohnnyaskingme,"Whyisn'titjustasgoodorbettertoaimwellfromthestartingpoint?"IremindedhimofGauss'sfamousworkonplanetaryorbits calculatedfromseveralobservations.Hequicklythoughtaboutthisforafewminutesandcametotheconclusionthatindeedthiswasasuperiormethod. IalsonoticedthatatmymentionofballisticprojectilessomepeoplemadeembarrassednoisesandIguessedthattherewasalreadysomeworkgoingononthis. Peoplewouldnotdiscloseeverythingthatwasbeingworkedonandclearanceswerenotuniformamongtheparticipants.Thisbringsmetoapointaboutvon Neumannthatseemstohavepuzzledmanypeople.Itconcernshisrelationswiththemilitary.Heseemedtoadmiregeneralsandadmiralsandgotalongwellwith them.Evenbeforehebecameanofficialhimself,anAECcommissioner,hespentanincreasingamountoftimeinconsultationwiththemilitaryestablishment.OnceI askedhim:"Howisit,Johnny,thatyouseemtobesoimpressedbyevenrelativelyminorofficerswhosometimesarenotsoveryremarkable?"AndIadded,inorder tosaysomethingalittlederogatoryaboutmyselftoo,thatwhatimpressedmemoreweresymbolsofwealthandinfluence,likethesightofJ.P.Morganmarchinginan alumniprocessionattheHarvardCentennialCeremoniesin

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1936.Ihadseenverymanywonderfulandeminentscientistsandartistsinmylifebefore,butthesightofthismanwhowasabillionaireandwieldedenormouspower reallyawedme.ButtogobacktoJohnny'sfascinationwiththemilitary,Ibelieveitwasduemoregenerallytohisadmirationforpeoplewhohadpower.Thisisnot uncommonwiththosewhoselifeisspentincontemplation.Atanyrate,itwasclearthatheadmiredpeoplewhocouldinfluenceevents.Inaddition,beingsofthearted, Ithinkhehadahiddenadmirationforpeopleororganizationsthatcouldbetoughandruthless.Heappreciatedorevenenviedthosewhoatmeetingscouldactor presenttheirviewsinawaytoinfluencenotonlyothers'thoughts,butconcretedecision­making.Hehimselfwasnotaverystrongoractivedebaterincommittee meetings,yieldingtothosewhoinsistedmoreforcefully.Onthewholehepreferredtoavoidcontroversy. Thesewerethedaysofdefenseresearchcontracts.Evenmathematiciansfrequentlywererecipients.JohnnyandIcommentedonhowinsomeoftheirproposals scientistssometimesdescribedhowusefultheirintendedresearchwasforthenationalinterest,whereasinrealitytheyweremotivatedbybonafidescientificcuriosity andanurgetowriteafewpapers.Sometimestheutilitariangoalwasmainlyapretext.ThisremindedusofthestoryoftheJewwhowantedtoenterasynagogueon YomKippur.Inordertositinapewhehadtopayforhisseat,sohetriedtosneakinbytellingtheguardheonlywantedtotellMr.Bluminsidethathisgrandfather wasveryill.Buttheguardrefused,tellinghim:"Ganev,Siewollenbeten"["Youthief!Youreallywanttopray"].This,welikedtothink,wasaniceabstractillustration ofthepoint. Gamow,wholivedinWashington,wasaconsultantattheNavalResearchLaboratory.Oneofmyearlyso­calledbusinesstripstoWashingtoninvolvedaconsultation withhim.HeaskedmetotalkaboutMonteCarloandwediscussedmodelingland­battlesituations.Hewasinterestedinanddidalotofworkontankbattles.He usedMonteCarlo,for

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example,tosimulatelandscapes,whichhedubbedStanscapes. HelivedinthesuburbswithhisfirstwifeRhoandwouldsay,"LetusmeetatChebyshevCircle."Ofcourse,hemeantChevyChase.(ChebyshevwasaRussian mathematician,andthisishowhepronouncedit.)GraduallyheandRhohadincreasingmaritaldifficultiesandfinallyseparatedanddivorced.Hemovedtothespartan surroundingsoftheCosmosClub,wheretheonlygoodthingwastheprofusionofnewspapersandmagazinesavailabletothemembers.OnedayIreceivedasad letterfromhimsayinghewaslivingaloneandthatonhishousewasasignsaying,"ForSell[sic]." In1954theFermiswerespendingthesummerinEurope,partlyattheFrenchPhysicsInstituteinLesHouchesnearChamonix,partlyatVarennainItaly,wherethe EnricoFermiInstitutewasfoundedafterhisdeath.Itnowholdsconferencesoncurrenttopicsinhigh­energyphysicsandinparticlephysics,bothfieldsjustbeginning towardstheendofFermi'slife. IfIrecallcorrectlyFermihadappliedforaresearchgrantthatsummerandhadnotobtainedit,whichirritatedhimsomewhat.Thisseemedverystrangetome.Just liketheincredibleaffairofthegovernment'sniggardlycompensationfortheuseofhispatentonthemanufactureofisotopes.Hetoldmeoncethathebelievedheand hiscollaboratorswouldreceiveperhapssometenmilliondollarsfromthegovernment.WiththismoneytheywantedtoestablishafundforItalianstostudyinthe UnitedStates.Butatthattimetheystillhadnotreceived"aredcent,"ashesaid.Eventuallyasettlementwasmadebutitwassosmallthatitbarelycoveredthe lawyers'fees,ifIremembercorrectly. WearrangedtomeettheFermisinParis,wheretheyweretostayafewdays,andtodrivetogetherintwocarspartofthewaysouth.Theyplannedtorentasmall FiatbutthedirectorofFiatinParismadeagreatpointofgivingthemaveryspecialeight­speedcar.IrememberFermiin­

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vitingmetotrythecarwithallitsspeedchangesalongthequaisandtheRuedeRivoli. Enrico'shealthwasbynownotthebest.TheprecedingsummerinLosAlamos,hiswifeLaurahadnoticedthathisappetitewaspoor,andthisbegantoworryher.He alsoshowedlessenergyforexerciseandinthetennisgameshelovedtoplay.Buttherewerenootherphysicalsymptoms,andLaurathoughtthatthiswaspossibly duetohisinvolvementintheH­bombcontroversyandtheOppenheimeraffair,andtohisskepticismandpessimismaboutthegeneralstateofworldpolitics.She hopedthesummer'srestawayfromhomewoulddohimgood. TheFermisalwayslivedsimplyandfrugally,andinPariswenoticedthattheywerereluctanttofrequentthe"good"andexpensiveFrenchrestaurants.Enricodidnot reallyenjoyfoodthatsummer.Neithercouldwepersuadethemtostayinfirst­classhotels,whichtheycertainlycouldaffordbetterthanwecouldinthosedays.On ourlittleovernightjointtripwefollowedsuitandtooksimplelodgingswiththeminamodest,smallinnintheValléeduCousinsomehundredfiftymilessouthofParis. Itiscurioushowoneremembersphysicalsettings.Itwaslateevening,therewerestarsintheskyandwesatonaterracenexttoamurmuringstreamdiscussingthe OppenheimerAffair.SomeelectricalwireswerestrungbetweentwohousesandallthewhilewetalkedFermiwaslookingatabrightstar,andbymovinghisheadso thewireswouldhideitfromsight,hewasobservingthescintillation. WeagreedthattheaffairwouldultimatelyleadtoabeatificationofOppenheimer;hewouldbecomeagreatmartyrandhisaccuserswouldbedamned.BothFermi andvonNeumannwere,inthehearings,fullyonOppenheimer'ssideanddefendedhimagainsttheaccusations,thoughneitherwasagreatpersonalfriendorspecial admirerofhis.Fermiwasnotgreatlyimpressedbyhisphysicsandhadsomereservationsabouthispoliticalleanings.HefelthoweverthatOppenheimerhadbeen treatedveryshabbily.We

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alsodiscussedtheattitudeofEdwardTeller,andIaskedFermihowheviewedthefuture.Suddenlyhelookedatmeandsaid:"Idon'tknow,I'lllookatitfromup there,"pointingatthesky.Didhehavesomepremonitionthathewasdangerouslysick?Ifso,heneveradmitteditinsomanywordsnordidhelookit.Butthisstruck measaboltfromtheblue,especiallysinceherepeateditoncemoreaswediscussedthefoundationsofphysics,themysteriesofparticles,thebehaviorofmesons, andhischanginginterestsfromnuclearstructuretothesupposedlymorefundamentalpartsofthephysicsofparticles.Againhesaid:"I'llknowfromupthere."The nextdayweseparated,theFermisdrivingeasttoGrenobleandLesHouches,andwe,Claire,andFrançoise'sbrothersouth,tospendavacationinLaNapoule,near CannesontheFrenchRiviera. WhenwereturnedtotheUnitedStatesattheendofthesummer,thenewswasthatFermiwasveryill,thatanexploratoryoperationhadbeenperformedimmediately uponhisreturntoChicagoandthatageneralizedcanceroftheesophagusandstomachhadbeenfound.Someofhisfriendsthoughtthecancermighthavebeen causedbyhisearlyworkwithradioactivematerialsatatimewhenprecautionswerenotverycarefullyobserved.IwonderedthenwhetherahabitofhiswhichIhad noticedofoccasionallyswallowinghard,andwhichIthoughtwasadeliberateformofself­control,mighthavebeenconnectedallalongwithaphysicaldifficulty. Hisillnessprogressedrapidly.IwenttoChicagotovisithim.InthehospitalIfoundhimsittingupinbedwithtubesintheveinsofhisarms.Buthecouldtalk.Seeing me,hesmiledasIcameinandsaid:"Stan,thingsarecomingtoanend."Itisimpossibleformetodescribehowshatteringitwastohearthissentence.Itriedtokeep composed,madeafeebleattemptatajoke,thenforaboutanhourwetalkedaboutmanysubjects,andallalonghespokewithserenity,andunderthecircumstances reallyasuperhumancalm.HementionedthatTellerhadvisitedhimthepreviousday,and

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jokedthathehad"triedtosavehissoul."Normallyitisthepriestwhowantstosavethesoulofthedyingman;Fermiputittheotherwayround,alludingtothepublic hullabalooaboutTellerandtheH­bomb.Perhapstheirconversationhadaneffect,forshortlyafterFermidiedTellerpublishedanarticleentitled"TheWorkofMany People,"toningdowntheassertionsofShepleyandBlair.DuringmyvisittoFermiLauradroppedinandIwasamazedattheordinarynatureoftheirconversation aboutsomehouseholdappliance. WetalkedonandIrememberhissayingthathebelievedhehadalreadydoneabouttwo­thirdsofhislife'swork,nomatterhowlonghemighthavelived.Headded thatheregrettedalittlenothavingbecomemoreinvolvedinpublicaffairs.Itwasverystrangetohearhimevaluatinghisownactivity—fromtheoutside,asitwere. AgainIfeltthatheachievedthissuperobjectivitythroughsheerwillpower. Somehowtheconversationturnedtotheprogressofmedicine.Hesaid,"WellStan,youknow,mychanceoflivingthroughthisisperhapsnotzerobutitislessthan oneinahundred."Ilookedathimquestioninglyandhecontinued,"Ibelievethatintwentyyearsorsoachemicalcureforcancerwillbefound.NowIhaveonlytwo orthreemonths,andassuminguniformprobabilities,theratioofthesetimesisonehundredtoone."Thiswashischaracteristicwayoftryingtobequantitative,evenin situationswhereitisnotpossible.ThenhalfseriouslyIraisedthequestionwhetherinathousandyearssomuchprogresswillbemadethatitmaybepossibleto reconstructpeoplewhohadlivedearlierbytracingthegenesofthedescendants,collectingallthecharacteristicsthatmaketipapersonandreconstructingthem physically.Fermiagreed,butheadded:"Howaboutthememory?Howwilltheyputbackinthebrainallthememorieswhicharethemakeupofanygivenindividual?" Thisdiscussionnowseemsratherunrealandevenweird,anditwaspartlymyfaulttohaveputuson

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suchasubject,butatthetimeitcamequitenaturallyfromhissuper­detachmentabouthimselfanddeath.Ipaidhimonemorevisit,thistimewithMetropolis;whenwe cameoutofhisroomIwasmovedtotears.OnlyPlato'saccountofthedeathofSocratescouldapplytothescene,andparaphrasingsomeofthewordsofKritoI toldNick,"Thatnowwasthedeathofoneofthewisestmenknown." Fermidiedshortlyafter.AshorttimelaterIwaspassingthroughChicagoagainandcalledonLaura.Igavetheaddresstothedriverandaddedthatthiswasthe houseofthewidowofthefamousItalianscientistwhohadjustdied.Thedriver,whohappenedtobeItalianandwhohadreadaboutitinthepapers,absolutely refusedtoletmepaythefare.OnlywhenItoldhimhecouldgivethemoneytocharitydidhetakeit. JustafterJohnnywasofferedthepostofAECCommissionerandbeforeheacceptedandbecameonein1954wehadalongconversation.Hehadprofound reservationsabouthisacceptancebecauseoftheramificationsoftheOppenheimerAffair.HeknewthatthemajorityofscientistsdidnotlikeAdmiralStrauss'sactions anddidnotsharetheextremeviewsofTeller.SomeofthemoreliberalmembersofthescientificcommunitydidnotlikeJohnny'spragmaticandratherpro­military viewsnordidtheyappreciatehisassociationwiththeatomicenergyworkingeneralandwithLosAlamosinparticular,especiallyhiscontributionstotheworkonthe AandHbombs.HerecognizedthisfeelingevenamongsomeofhisPrincetonassociates,andhewasafraidthatitwouldbecomestrongerwhenhejoinedtheAtomic EnergyCommission.ThisdespitethefactthatintheOppenheimerAffair,eventhoughhedidnotespeciallylikeOppenheimerpersonally,hedefendedhimwithgreat objectivityandgaveverycorrect,courageous,andintelligenttestimony. ThedecisiontojointheAEChadcausedJohnnymanysleeplessnights,hesaid,andinatwo­hourvisittoFrijolesCanyononeafternoonhebaredhisdoubtsand askedme

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howIfeltaboutit.Hejoked,"I'llbecomeacommissionnaire."(InFrenchthetermisusedtomeanerrandboy.)Buthewasflatteredandproudthatalthoughforeign bornhewouldbeentrustedwithahighgovernmentalpositionofgreatpotentialinfluenceindirectinglargeareasoftechnologyandscience.Heknewthiscouldbean activityofgreatnationalimportance.Indeed,withhissupremeintelligence,hecouldhavedoneanenormousamountofgoodinseeingwhatwasvaluableincertain programsandininitiatingnewones.Asafriendofhisandhavingpressedhimtoaccepttheoffer,Strausswouldbeobligatedtosupporthisviewsandideas.Besides, JohnnyhadabitoftheTeutonictraitofbeingeasilyimpressedbyofficialdom.Atanyrate,hewastornbetweentwopoles:afeelingofpridewiththehopeofdoing somethinggoodandusefulandthefearofbecomingassociatedinthemindsofhiscolleagueswithasmallminorityofthescientificcommunityandofcareer­oriented persons.AcceptancerequiredtakingaleaveofabsencefromtheInstituteandsomefinancialsacrificeaswell.IdonotknowthedetailsofthepromisesthatStrauss mayhavemadeorthepressureshemayhaveexerted. Iwonderedlaterwhetherthisdecisionandtheanguishandnervoustensionitcausedhimhadperhapspredisposedhimtotheonsetofthefatalillnessthatcamenot longafter.Obviouslytakingthisstepdidhavesomephysicalimpact,forhelookedwanandshowedtheeffectsofstress.Besides,therewasalotmorephysicalwork involved.Idon'tthinkthateverbeforehehadworkedfromeightinthemorningtofiveintheafternooninthesameplaceandwithseveralmeetingseveryday.No matterhowhardhehadworkedbeforeithadbeenonhisowntimeandchoosing.Thefirstindicationthatsomethingwasverywrongwithhishealthappearedsome timeafterhebecameacommissioner. AsIhadknownhimovertheyearshealwaysseemedingoodhealth.Heonlyhadinfrequentcolds,sleptwell,workedhard,couldeatanddrinkliberallywithout showinganyeffects.Idon'tthinkhewashypochondriac.Onthecon­

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trary,exceptforanoccasionalcoldortoothache,hewasverylittlepreoccupiedbyhisownphysicalstate,althoughheonceshowedmesomecorrespondencehehad hadwithDr.JanosPleschaboutkidneyfunction.OnceononeofourmanywalksinFrijolesCanyonheremarkedinpassingatreethrottledbyavine,howhorribleit mustfeeltobesurroundedandtrappedandunabletogetaway.Thisremarkcamebacktomymindlaterwhenhebecameparalyzed. Igotwindofsomevaguerumorsthathewasill.IaskedTelleraboutit,buthegavemeevasiveanswersandsaidsomethingIcouldnotinterpret.Itelephonedthe houseinGeorgetown,andKlaritoldmeanoncommittallittlestory.Icouldnothelpbutsuspectthatsomethingwasverywrong,andlaterfoundoutthatJohnnyhad givenspecificordersthatIshouldnotbetoldthathehaddevelopedcancer.Onedaysittinginhisofficehehadbeenseizedbyaviolentpainintheshoulder,sostrong thathealmostfainted.Thispaindisappeared,buthewenttoMassachusettsGeneralHospitalinBostonwhereasmallcancerousgrowthwasremovedfromhis clavicle,probablyalreadyasecondarygrowth.HesoonrecoveredfromthissurgicalinterventionandcametoLosAlamosforwhatwastobehislastvisitthere.Istill hadnotbeentoldwhatwaswrongwithhim. HecametoourhouseandInoticedthathelimpedslightly.Heseemedobviouslypreoccupiedandperturbed.Therewasasadnessinhimandhefrequentlyseemedto lookaround,asif,itoccurredtomelater,hemighthavebeenthinkingthatthiswasperhapshislastvisitandhewantedtorememberthescenery,themountains,the placesheknewsowellandwherehehadsooftenhadinterestingandpleasanttimes.YetatthesametimehejokedabouthispresenceinLosAlamosasa commissioner.Nowhewastherenotonlytothinkaboutscientificmattersbutaboutveryprosaicadministrativeones.AndRabiwhowasintownalsochimedinthatit wasnolongerascientificvisitbutaninspectiontour.Beforeheleft,FrançoiseshowedhimarecentsnapshotofClaireonherbicycle.Heaskedifhecould

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takethephotowithhim.HewalkedbacktotheLodgethroughthegarden,andwatchinghimthroughthewindowIdefinitelyhadthefeelingthatsomberand melancholythoughtswereinhishead. AfewweekslaterononeofmyvisitstoWashingtonJohnnytookmetolunch.Duringthemealhetoldmethatdoctorshaddiscoveredhehadcancerandhe describedwhatkind.Thiswasatremendousshockforme.ItoldhimmysuspicionsthatsomethingwaswrongandthatforsomereasonIhadwonderedifitcouldbe diabetesorhisheart.IturnedawaysoasnottoshowhowupsetIwas,buthenoticeditanywayandstartedtellingajokeaboutawomaninBudapestwhosemaid hadfallenill.Shesentforadoctorwhotoldherthathermaidhadsyphilis.''ThankGod,"thewomansaid,"Iwasafraiditwasmeaslesandshewouldinfectthe children."Duringthisdramaticlunchhestillshowedgreatstrengthofwillandnosignsofknucklingunder.Iwasshatteredandwonderedwhetherhewouldever recover. OnmynexttripIvisitedJohnnyathome.TheGeorgetownhousehehadrentedwasverydifferentfromhisPrincetonone.Itwassmall,veryseventeenth­century Dutchwithablack­and­white­tiledvestibuleasinsomeofVermeer'spaintings.HewasstillworkingattheAECbutwalkedwithincreasingdifficultyandsoonhadto taketoawheelchair.Friendsandevendoctorswonderedwhethersomeofitwaspsychosomatic.Itwasneverclearwhatkindofcancerheactuallyhad.Inever learnedthewholestory;Idon'tthinkmanypeopleknewit.Klariwouldneversaymuchaboutit.Iwastolditstartedintheprostateandultimatelymetastasizedsothat hebecamepartlyparalyzed. DuringhisillnessJohnnydidnottalktomeabouthisimportantworkontheICBMCommittee;onlylaterIlearnedthathewasChairmanandthatitwascalledthevon NeumannCommittee.Ashegrewincreasinglyill,someofthemeetingswereheldinhishouseandlaterattheWoodnerHoteltowhichthevonNeumannshadmoved so

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astobeclosertoWalterReedHospital,wherehewasundergoingtreatment.Totheendhemaintainedthiscompletediscretion.EventhoughIwasperhapsoneofhis closestfriendsheneverbroachedtomeclassifiedormilitarysubjectsinwhichIwasnotinvolved.Ourusualconversationswereeitheraboutmathematicsorabouthis newinterestinatheoryofautomata.Theseconversationshadstartedinasporadicandsuperficialwaybeforethewaratatimewhensuchsubjectshardlyexisted. Afterthewarandbeforehisillnessweheldmanydiscussionsontheseproblems.Iproposedtohimsomeofmyownideasaboutautomataconsistingofcellsina crystal­likearrangement.ThismodelisdescribedinthebookeditedbyArthurBurks,CellularAutomata,andinBurks'sownbookonthetheoryofautomata.At thattimeitwasbelievedthattherewere1010neuronsinthesmallspaceofthehumanskullandthatfromsomeneuronsthereissuedsomehundredsandperhapsinthe centerregionathousandconnectionswithotherneurons.Weusedtomarvelatthecomplicationoftheorganizationofthebrain.NowIunderstandthatithasbeen foundthattherearethousandsofconnectionsfromeachneurontoothersandinsomeareasfiftythousandandevenmore.Andeachneuron,whichatthattimewas believedtobejustarathersimple"flipflop.""yesorno"machine,isnowbelievedtobeacomplicatedorganwithmanymorefunctions.Inthespaceoffifteenyears, sincevonNeumann'sdeath,thefactshavebecomeamplified;thewholestructureisevenmoreamazing,moreincrediblethanitseemedatthetime.Johnnydidnotlive toseethedevelopmentsfollowingCrickandWatson'sworkonthestructureoftheDNAchainsinthenucleusofcellsandthecodewhichtheycontain. ItisevidentthatJohnny'sideasonafuturetheoryofautomataandorganismshadrootsthatwentbackintime,buthismoreconcreteideasdevelopedafterhis involvementwithelectronicmachines.Ithinkthatoneofhismotivesforpressingforthedevelopmentofelectroniccomputerswas

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hisfascinationwiththeworkingofthenervoussystemandtheorganizationofthebrainitself.Afterhisdeathsomeofhiscollaboratorscollectedhiswritingsonthe outlinesofthetheoryofautomata.Publishedposthumously,hisbookonthebrainhadmerelythebarestsketchesofwhatheplannedtothinkabout.Hediedso prematurely,seeingthepromisedlandbuthardlyenteringit.Thegreatdevelopmentsinmolecularbiologyreallycametoolateforhimtolearnmuchaboutitandto enterafieldwhichIknowfascinatedhim. Anothersourceofstimulationcamefromhisinterestinthetheoryofgames.Thiswasinitiallyperhapsanindependentcuriosity,butinmyopinion,ageneraltheoryof contests,fightsforsurvival,andevolutionwillfurnishinthenearfutureawholeclassofnewmathematicalproblemsandnewpatternsofthoughtconcerningthe schemataofthedevelopmentofbiologicalprocessesthroughwhatisnowcalledevolutionaryand"survivalofthefittest"competition.Inthatareaoneofhismajor undertakingswastheelaborationandcreationofnewmodelsofprobabilistictheoryofgames,inparticularthestudyoftherulesofcoalitions.Hedevelopedthese ideaswithOskarMorgenstern,aPrincetoneconomist,inamonumentalbookentitledTheoryofGamesandEconomicBehavior. InthespaceoffifteenyearssincevonNeumann'sdeaththenewfactsdiscoveredhavebecomemoreperplexing,thewholestructureisevenmoreamazing,more incrediblethanitseemedatthetime.Itwillgoonincreasingasourunderstandingofanatomyandphysiologyimprovesandwillleadtonewfieldsofmathematical research. Thisprocessofincreasingcomplexityinscienceisgoingonwithnosignasyetofslowingdown.Whetheritwillcontinueindefinitelyorregressisabigquestion.Itisa partoftheproblemofinfinityversusthefinitenessoftheworld. Inthelastmonthsofhislife,JohnnywashospitalizedatWalterReedHospital.Heoccupiedaverylargesuitereservedforhighgovernmentofficials.Inthefallof1956 wewerelivinginCambridgeagain,andIwasavisitingprofes­

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soratMITonanotherleavefromLosAlamos.ImanagedtotraveltoWashingtonandvisithimafewtimes.Ononeofthesevisitsonceagainwehadadiscussion aboutage.Hewonderedhowmuchmoreoriginalandcreativeworkhecouldstilldoifhelived.Itriedtoencouragehimbytellinghimthathestillcoulddoatleasthalf asmuchagain. Curiously,threeyearsearlierwhilevisitingFermiinhishospitalinChicago,ourconversationhadalsoturnedtothesametopic;Fermihadsaidcalmlythathe consideredhehadalreadydonemostofhiswork.Whatadifferenceinoutlook,oratleastinthewaythesetwogreatmenexpressedorsuppressedtheirfeelings. OnthesamevisitIwentbymistaketotheoppositecornerofthehospitalbutonthesamefloor,andwalkedintoanantechamberwheretwomilitarymenweresitting. Theylookedatmeinsurpriseandquestioningly.IsaidIwastheretovisitafriendandtheirlookturnedincredulous.WhenIadded,"Dr.vonNeumann,"theysmiled anddirectedmetotheproperrooms.IhadenteredthePresidentialSuitewherePresidentEisenhoweratthatmomentwashospitalizedafterhisheartattack.Itold thistoJohnnywhenIregainedhisroom.Heenjoyedthis.ItamusedhimtobeinalocationsymmetricallyoppositetothatofthePresidentoftheUnitedStates. Somemonthsbefore,AdmiralStrausshadaconversationwithmeaboutwhatJohnny'slifecouldbeshouldherecoversufficientlytoleavethehospitalbutnot sufficientlytorejointheCommission.Theideawastocheerhimupwithnewsurroundingsandperhapsprovidehimwithaperspectiveonthingsotherthan governmentalwork.Strauss,thoughnotbelievingthatafullrecoverywouldcometopass,wasinstrumentalinobtainingforhimanofferofaspecialprofessorshipat UCLA.ThisprospectdivertedandcheeredJohnnysomewhat. Henevercomplainedaboutpain,butthechangeinhisattitude,hisutterances,hisrelationswithKlari,infacthiswholemoodattheendofhislifewereheartbreaking. At

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onepointhebecameastrictCatholic.ABenedictinemonkvisitedandtalkedtohim.LaterheaskedforaJesuit.Itwasobviousthattherewasagreatgapbetween whathewoulddiscussverballyandlogicallywithothers,andwhathisinnerthoughtsandworriesabouthimselfwere.Itwasvisibleonhisface.Johnnyusedtobe completelyagnosticeventhoughhesometimesexpressedhisfeelingsofwonderandmystery.OnceinmypresencewhenKlarichidedhimforhisgreatself­ confidenceandprideinhisintellectualachievements,herepliedthatonthecontraryhewasfullofadmirationforthewondersofnatureandtheevolutionofthebrain, comparedtowhichallwedoispunyandinsignificant. Bythenhewasvery,veryill.Iwouldsitwithhimandtrytodistracthim.Therewasstillsomescientificcuriosityinhim;hismemorystillseemedtoworksporadically, andonoccasionalmostuncannilywell.Iwillneverforgetthesceneafewdaysbeforehedied.IwasreadingtohiminGreekfromhisworncopyofThucydidesa storyhelikedespeciallyabouttheAthenians'attackonMelos,andalsothespeechofPericles.Herememberedenoughtocorrectanoccasionalmistakeor mispronunciationonmypart. JohnnydiedinWalterReedhospital,February8,1957.HewasburiedinPrincetoninabriefCatholicservicewithashorteulogybyAdmiralStrauss.Afterthe funeraltherewasasmallgatheringathishouse.Severalmathematicianswerethere,amongthemhisoldfriendJamesAlexander,himselfrecoveredfromanillnessnot unliketheoneIhadhadinLosAngeles.AlsopresentwereAtleSelberg,thenumbertheorist,andLewisDelSasso,anengineerwhohadworkedatbuildingthe MANIAC,andMrs.Gorman,hislong­timesecretaryattheInstitute.Afterhisdeath,FrançoisewenttoWashingtontospendafewdayswithKlari,takingClaire along.ThepresenceofachildshewasfondofhelpedmomentarilytotakeKlari'smindoffthelonggruelingmonthsthatprecededJohnny'sdeath. VonNeumannwasremarkablyuniversal.Ihaveknown

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wonderfulmathematicianswhowereseverelylimitedintheircuriosityaboutothersciencesbuthewasnot. VonNeumann'sreputationandfameasamathematicianandasascientisthavegrownsteadilysincehisdeath.Morethanhisdirectinfluenceonmathematical research,thebreadthofhisinterestsandofhisscientificundertakings,hispersonalityandhisfantasticbrainarebecomingalmostlegendary.True,inhislifetimehehad alreadyachievedanenormousreputation,andallthehonorsthemathematicalworldcangive.Buthehadhisdetractors.Hewasnotentirelywhatonemightcalla mathematicians'mathematician.Puristsobjectedtohisinterestsoutsideofpuremathematicswhen,veryearly,heleanedtowardsapplicationsofmathematicsorwhen hewrote,asayoungman,aboutproblemsofquantumtheory. Asformyself,IwasnevergreatlyimpressedbyhisworkonHilbertspaceoroncontinuousgeometry.Thisisaquestionoftaste,andwhenIwasmoreofapurist myselfImadegood­naturedfunofcertainofhisinvolvementsinapplications.Itoldhimonce,"Whenitcomestotheapplicationsofmathematicstodentistry,maybe you'llstop." Buttherewasnothingsmallabouthisinterests,andhisexquisitesenseofhumorpreventedhimfromgoingoffontangentsfromthemainedificeofmathematics.He wasuniqueinthisrespect.Unique,too,werehisoverallintelligence,breadthofinterest,andabsolutefeelingforthedifferencebetweenthemomentarytechnicalwork andthegreatlinesofthelifeofthemathematicaltreeitselfanditsroleinhumanthought. NowBanach,Fermi,vonNeumannweredead—thethreegreatmenwhoseintellectshadimpressedmethemost.Theseweresadtimesindeed.

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PARTIV THEPASTFIFTEENYEARS

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Chapter13 GovernmentScience 1957–1967 Itismoredifficulttowriteaboutrecentevents;theperspectiveispoorer,theseparationofthecharacteristicortheimportantfromthefortuitousismoredifficult.My storyofthepastfifteenyearsorsowillthereforebecontractedandwillconcernactivitiesandpeopleevenmorearbitrarilychosenthanthereminiscencesand reflectionsoftheearlierchapters. ReturningtoLosAlamosin1957afterayear'sleaveofabsenceatMIT,IwasaskedbyBradburytoacceptoneofthetwonewlycreatedpositionsofresearch advisortothedirectorofthelaboratory.TheotheradvisorwastobeJohnManley,aphysicistwhohadheldimportantadministrativepostsatLosAlamosduringthe warandwantedtoreturntoNewMexicoafteralongabsenceasprofessorattheUniversityofWashingtoninSeattle.Administratively,researchadvisorsweretobe onthesamelevelasdivisionleaders,andtheirdutiesweretooverseetheresearchactivitiesthrough­

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outthelaboratory,inthevariousdivisions:theoretical,physics,chemistryandmetallurgy,weapons,health,"Rover"(nuclearrocket),andothers.Togetherwetriedto influencethevariousprogramsofthelab.Thiswasanarduousandmany­sidedtask,andtalkingwithmanypeopleabouttheirresearchactivitiesenabledmeto broadenmyowninterests.Iheldthispositionuntil1967whenIretiredfromLosAlamosandjoinedthemathematicsdepartmentattheUniversityofColoradoin Boulder.Inlaboratoryjargon,ManleyandIwererespectivelyknownasRAJMandRASU. InmyadministrativeroleinLosAlamosandlateraschairmanofthemathematicsdepartmentinBoulder,Icametounderstandandappreciatebetterandalso commiseratewithfriendsandacquaintanceswhohadbecomefullyoccupiedbyadministrativeduties.InmyyoungeryearsIhadhadtheusualskepticalattitude towardsmostchairmenofdepartments,deans,presidents,directors,andthelike.Therehadbeenexceptions,ofcourse.OneofthesewasJ.CarsonMark,leaderof thetheoreticaldivisioninLosAlamossincethemiddleforties.MarkisaCanadianmathematicianwhocametoLosAlamostowardstheendofthewarwiththe BritishMission.HebecameanAmericancitizensomeyearslater.HeborethebruntofthedifficultieswithTellerwithremarkablecalmandobjectivity;heisoneofthe fewmathematiciansIknowwhohaveanunderstandingfortheproblemsofphysicsandassociatedtechnologyinabroadsense.Hisdirectionofthetheoretical divisionwasanexampleofintelligentmanagementofascientificgroupwithoutexercisingunduepressuresforprogrammaticwork.Hewasabletoencouragefree scientificpursuitsinareaswhichwereonlyindirectlyrelatedtothetasksofthelaboratory,andhesupportedtheoreticalphysicsandappliedmathematicsinthebest sense.(Incidentally,hewasalsoaregularparticipantofourpokersessions.From1945untilnowIcannotrecallasingleoccasionwhenherefusedtocomeormissed oneofourgames.Thesebynowhaveconsiderablyslowedinfrequency.Fromweekly

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theybecamemonthlyandnowoccuronlyoccasionally,mainlywhenIhappentobevisitingLosAlamos.) AfterthewarithadbecomeclearthatscienceandtechnologyhadbecomesocrucialfornationalaffairsthatthegovernmentsoftheWesternworldhadtodevote enormousamountsoftimeandhugebudgetstothem.Famousscientistswerecalledupontoentertheinnercirclesofgovernmenttohelpdirecttheircountries' scientificactivities,notonlyforthearmsracebutfortechnologicaladvancement.ChurchillhadhadLordCherwell;DeGaulle,FrancisPerrin;America,herScientific AdvisoryCommittees.BeginningwithBushandConant,Oppenheimer,vonNeumann,andmanyothersbecamegovernment"sages."Governmentsciencepeaked andcommitteesproliferatedundertheEisenhower,Kennedy,andJohnsonadministrations,andevenIfoundmyselfcalledupon.UntilthenIhadalwaysresistedbeing drawnintoanykindoforganizationalposition;foryearsIcouldclaimthatmyonlyadministrativejobhadbeenontheWineTastingCommitteeoftheSocietyof FellowsbackinmyearlyHarvarddays. AfewyearsbeforeJohnny'sdeathandincreasinglysoafter,asaresultofmyworkonthehydrogenbomb,Ibecamedrawnintoamazeofinvolvements.Thesehad todopreciselywithgovernmentscienceandwithworkasamemberofvariousSpaceandAirForcecommittees.Also,insomecirclesIbecameregardedasTeller's opponent,andIsuspectIwasconsultedasasortofcounterweight.SomeofthesepoliticalactivitiesincludedmystandontheTestBanTreatyandtestimonyin Washingtononthatsubject.ThecartoonistHerblockdrewintheWashingtonPostapictureoftherespectivepositionsofTellerandmeinwhichIfortunately appearedasthe"goodguy." SinceIneverhavekeptnotesordiariesofanykind,Imaynotalwaysbeentirelycorrectaboutthechronologyofeventsorhowthingsandpeoplewereconnected duringthesebusyyearsofscientifico­technologicalactivities. Washingtoncommittees,Isoonnoticed,wereoftenvery

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enviousofnewideas,withtheirmembersexhibitingthewell­known''notinventedhere"syndromeofrejectingproposalsorideasonlybecauseofthevestedinterests ofcommitteemembers.Thisfeelingwasagreaterobstacletothedevelopmentofnewprojectsthantheconcernabouttheircostandtheamountsofmoneythey wouldrequire.Decisionsalsoseemedsometimesdictated,notsomuchbyobjectiveevaluationasbytheusualacademicrivalriesandenvyofscientificfame.HadI notbeenalreadyquiteoldandcynicalthiswouldhavemademeleavegovernmentalsciencealtogether.IrememberedhowinLosAlamosJohnnyremarkedon severaloccasionsthatitwasnoteasytointroducenewthings;onehadtopersuadeeveryjanitor,hesaid.Butoncesomethingwasaccepteditbecameasortofbible anditwasequallyhardorevenimpossibletochangeitorgetridofit.Thisnationalsituationhasbecomeevenworsenow,partlybecauseoftherecentspreading skepticismregardingthevalueofscienceanditsbenefits,andakindofgeneralpassivitysoremovedfromthetraditionaltraitsofAmericanenterprise,energy,and spiritofcooperation. Theideaofnuclearpropulsionofspacevehicleswasbornassoonasnuclearenergybecameareality.Itwasanobviousthoughttotrytouseitsmorepowerful concentrationofenergytopropelvehicleswithaverylargepayloadforambitiousspacevoyagesofexplorationorevenforexcursionstothemoon.IthinkFeynman wasthefirstinLosAlamosduringthewartotalkaboutusinganatomicreactorwhichwouldheathydrogenandexpelthegasathighvelocity.Asimplecalculation showsthatthiswouldbemoreefficientthanexpellingtheproductsofchemicalreactions. Ibecameinvolvedwithtwosuchprojects,oneinanadvisorycapacity,withtheothermoredirectly.ThefirstwasProjectRover,anuclear­reactorrocketwhichwas beingdesignedinLosAlamosalreadyquiteafewyearsbeforetheRussianSputnik,butwithverylimitedfunds.Thesecond

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wasaspacevehicle,laternamedOrion.Around1955EverettandIwroteapaperaboutaspacevehiclepropelledbysuccessiveexplosionsofsmallnuclearcharges. TheideahasevenbeenpatentedbytheAECinournames.ThismethodcouldbemuchmorepowerfulthanRoverandisaveryambitiousbutefficientwayto undertakespaceexplorationswithavehicleabletotravelathighspeedswithhighpayloadsandanextremelygoodratioofpayloadtototalinitialweight.The spaceshipcouldtransporthundredsorthousandsofpeople.WhenKistiakowskiwasPresidentEisenhower'sScientificAdvisorIinformedhimaboutsuch possibilities,buthisreceptionofitwasnotenthusiastic.ButmoreaboutOrionlater. SoonafterJohnKennedy'selectioninNovember1960,IreceivedatelephonecallfromJerryWiesnerfromCambridge.IhadmetWiesnertheyearIwasatMITas avisitingprofessor;wehadseeneachotherseveraltimesandhadhadgoodconversationsaboutscienceprojects,nationalprograms,education,andsoon.Wehad talkedalsoabouttheTellerbusiness.WiesnerwaswaryoftheEdwardianbrandofpolitics.IwasnottoosurprisedwhenIreceivedthiscall.Itmusthavelastedmore thanhalfanhourandJerryinformedmethatPresidentKennedyhadappointedhimchairmanofataskforceonscienceandtechnology.Heaskedwhatmyideas wereaboutthenationallyimportantscientificortechnologicalprojectsthePresidentshouldknowaboutandconsiderforthecountry."Howaboutgoingtothemoon?" Iasked.Iimaginedozensofotherpeoplehadmadethesamekindofsuggestion.InhisinauguraladdressKennedyproposedanationalprojecttoputamanonthe moon.Myinvolvementwiththespaceeffortbeganinearnestwiththatconversation.IbecameconsultanttoWiesner'sScientificAdvisoryCommitteeandvisited Washingtonfrequently. Immediatelyafterthewar,ClintonP.Anderson,SenatorfromNewMexico,aformermemberofPresidentTruman'scabinet,becameoneofthemostinterested, knowledgeable,

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influential,andeffectiveproponentsoftheusesofnuclearenergy.HewasinstrumentalinhelpingtheLosAlamosLaboratoryandtheassociatedbiginstallationin Albuquerque,theSandiaLaboratory. IbecameacquaintedwithhimduringoneofhisearlyvisitstoLosAlamos,enjoyinghisconfidenceand—itseemstome—histrustinandrelianceonmyopinions,not onlyintheareaofnuclearenergybutalsointhefieldofspaceactivities.SeveraltimesheinvitedmetotestifybeforeCongressonspecificspacematters,suchasthe organizationofNASAandwhetheritshouldbepartofthemilitaryestablishmentoranindependentorganization. WhenitwasdecidedtodosomethinginearnestaboutProjectRover,WiesnernamedaPresidentialCommitteetolookintothematter.Iwasoneofitsmembers. AmongsomememberswhowerechemistsInoticedadegreeofskepticismaboutitsworthandfeasibility,againmotivatedinmyopinionbytheirapprehensionthatit mightcompetewiththealreadyexistingchemicalrocketpropulsionsystemswhichwerebeingdeveloped.Someofthediscussionsremindedmeofthebigdebatesat thebeginningofthecenturybetweenlighter­than­airandheavier­than­airadvocates,orevenoftheearliercompetitionofsteamshipsversussailingvessels.Andindeed thecommitteewroteareportwhichbyfaintpraise,essentiallycondemnedProjectRovertoadefactodeathbyproposingtomakeitapurelytheoreticalstudy withoutfundsforexperimentalworkoranyinvestmentinconstruction.ThephysicistBerndMatthiaswastheonlymemberofthecommitteewhojoinedmeinwritinga dissentingopinion. SenatorAndersonwaschairmanofthecongressionalSpaceCommittee.HeknewmypositiononRover.Withhisfeelingforthepsychologicalandpolitical motivationsofcommittees,andhisvitalinterestinthenewtechnologyanditsimportanceforthenation,hetookmeonedaytotheofficeofthethenVice­President Johnson.TogetherwewalkedtoanearbybuildingtoseeWiesner.SinceJerryand

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Iwerefriends,itembarrassedmealittletobepresentatameetinginwhichJohnsonandAndersonwerepressinghimhardtochangetheattitudesoftheScientific AdvisoryCommitteeonquestionsofnuclearpropulsion.Theysupportedmyviewsinoppositiontohis.Ultimatelytheminorityopinionprevailed,andtheRover Projectwassaved.FundswereallottedforLosAlamosworkandovertheyearsitbecameanextremelysuccessfulventure.Unfortunatelyitwasstoppedagainlater byeconomiesinthespaceprogram. IwasalsoinvitedtojoinanAirForceCommitteeonasimilarsubject:generalproblemsofplansforspaceandtheAirForce'sroleinit.TheCommitteewaschaired byTrevorGardner,aformerassistantsecretaryoftheAirForceduringEisenhower'spresidency.GardnerwasaveryinterestingpersonofwhomIbecameveryfond. Hisvigorousandlustypersonality,hisgreatenergy,thewidescopeofhisimaginationappealedtomeverymuch.Ifoundhimverycongenial. Thecommitteeoriginallycomprisedanumberofpersonsimportantinscienceandtechnology.Theonlyothermathematician,MarkKac,waspresentatafew sessions.Amongthe"bigshot"members,IrememberHaroldBrown,directoroftheLivermoreLaboratory,laterSecretaryoftheAirForce,CharlieTownes,who receivedtheNobelPrizefortheinventionofmasers,GeneralBernardShriever,afrequentvisitor.VinceFord,anAirForcecolonelwhohadbeenJohnny'saideon thevonNeumannICBMCommittee,wasnowGardner'sassistant.HeorganizedthemeetingsofaworkingsubcommitteewhichmetinLosAlamos.Thesemeetings involvedmanypeoplefromthenewlybornaerospaceindustry.SometimeswemetinLosAngeleswheretheheadquartersoftheballisticsdivisionoftheAirForce underGeneralShrieverwerelocated.AtothertimeswemetinWashingtonwhereGeneralShriever,Gardner,FordandIdiscussedamongourselvesduringrestaurant luncheshowtoplantheexplorationofspaceandmoregenerallytheproblemsofspacestudyfortheAirForce.

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Earlyatoneofthesemeetings,somebodyfromindustrypresentedplansforretrievingrocketengines,whichwouldsavemoneybymakingthemreusable.Thereal problem,asIandsomeotherssawit,wastodosomethingimportantinputtingupthesatellitesanddoitquicklyratherthantostartbysavingmoney.Alsoitseemed tomethatboosters,namelytheengines,wereasmallpartoftheoverallcost,andthatitwouldbeawkward,tosaytheleast,tostartbyreusingsecond­handengines, perhapsdamaged.Whentheproponentsdronedonabouttheirideas,showingtheirplotsandgraphs,IwhisperedtoGardner,"Thissoundstomelikeaproposalto usethesamecondomtwice."Heburstoutlaughingandsenttheremarkaroundthetableinrepeatedwhispers.PerhapsthisjokesavedtheUnitedStatessome millionsofdollarsinexpenditureforwhatwouldhavebeenpointlessandimpracticalworkatthetime. OrionwasalsodiscussedbytheGardnerCommittee.AtmysuggestionTedTaylorbecametheexecutivedirectorofagroupworkingonit.Startingin1957,Taylor developedtheOrionideaasareactiontotheRussianSputnik,ashesaid.HeassembledanimpressivegroupofbrightyoungmenattheGeneralAtomicLaboratory inLaJolla,Calif.ThephysicistFreemanDysonbecameveryinterestedandenthusiasticandtookaleaveofabsencefromthePrincetonInstitutetoworkforayear withTaylor.Afewyearslaterhewroteaneloquentarticledescribingtheprojectandhowitwasputonashelf.Itappearedunderthetitle"LaVieetMortd'Orion"in theParispaperLeMonde. SomehowtheGardnerCommitteereportgotlostatahighlevelintheWashingtonmaze.WiesnerdisagreedwithGardnerabouttheroleoftheAirForceinspace. SomebodyinWashingtonmanagedtoburythereport,andIdon'tthinkPresidentKennedyeversawit.Thewholethingisstillamysterytome.AftertheGardner CommitteefinisheditsworkitwassucceededbyanotheronecalledtheTwiningCommittee.Itsmembersincludedsomehawkishtypeslike

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TellerandDaveGriggs.GeneralDoolittleofTokyoairraidfamewasalsoamember. IbecameconnectedwithTrevorGardnerlaterinamoreprivatecapacity.HeaskedmetojointhescientificadvisoryboardoftheHyconCorporationinCalifornia whichheheaded.Thecompanymanufacturedhighlysecretmilitaryequipmentincludingspecialcameras.Fowler,Lauritsen,AlHill,aphysicistfromMIT,andJesse GreensteinthePalomarastronomer,weretheothermembersoftheboard.IlearnedthatWiesnerandhisgroupinHyconEastandGardnerinCaliforniahadhad someseriousdisagreementsaboutfinancialproblemsconcerningthecorporation,andapparentlyWiesnerandGardnerwerebarelyonspeakingterms. TosomeGardnerwasacontroversialpersonbecauseofhisquicktemperandhisstrongopinions.Hehadgreatpoliticalambitions(hewouldhavelikedtobecome SecretaryofDefense),buthewasatcrosspurposeswithsomemembersoftheKennedyAdministration.HediedofaheartattackshortlybeforeKennedy's assassination.ItwasGardnerwhohadestablishedthevonNeumannICBMCommittee.ThishadbeenofimmenseimportancefortheU.S.spaceeffort;Ithinkit reallygotitofftheground.ThemilitaryandnationalimportanceofthisandotherGardnerinitiativescanhardlybeexaggerated. Atthesametime,Iwascontinuingmyownwork.AfterFermi'sdeathPastaandIdecidedtocontinueexploratoryheuristicexperimentalworkonelectronic computersinmathematicalandphysicalproblems.Wefeltthatthecombinationofclassicalmechanicsandastronomyproblemslentitselftotwokindsofstudies:one, thebehavioroflargenumbersofparticles—callthemstars—inaclusterorgalaxy;theother,thehistoryofasinglemassofgasasitdevelopedfrominitialconditions bycontractionatfirst,perhapsgivingrisetoadoubleormultiplestar,thengeneratingmoreandmorenuclearreactions,exhaustingitsnuclearmaterial,andfinally perhapscollapsing.Manycalculationshavebeen

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donesinceonthislatterproblemoverthelasttwentyyears,changingthewholeunderstandingofastrophysicsandofthedevelopmentoftheuniverseasfaras individualstarsareconcerned. TheproblemofclustersofstarswasIthinkthefirststudyofthisnatureusingcomputers.Wetookagreatnumberofmasspointsrepresentingstarsinacluster.The ideawastoseewhatwouldhappeninthelong­rangetimescaleofthousandsofyearstothespherical­lookingclusterwhoseinitialconditionsimitatedtheactual motionsofsuchstars.Thiswasareallypioneeringcalculationshowingthatthissortofinvestigationwaspossible.Itgaveverycuriousandunexpectedobservationsin classicalmechanics,formationofsubgroups,andcontractions.Wemadeafilmofthesemotionsonanacceleratedscalewhichshowedtheseinterestingphenomena. ThisworkgaverisetootherstudiesofthissortatBerkeley,inFrance,andelsewhere. AnotherproblemwhichIattackedbutwhichisstillnotsolvedisanattempttoseewhatwillhappenwhenamassofgasofverylargedimensions,sayofthewhole solarsystem,atverylowdensityandhavinginitiallyamildamountofturbulencestartstocontract.Howwoulditcontract,andhowwoulditfinallyformastar?What isinterestingandtheactualpurposeoftheproblemwastoseewhetherandhowoftenitwouldformadouble,triple,ormultiplestar.Thereasonforthiscuriosityis thatmanystars,inourneighborhoodatleast,aredouble.Accordingtorecentstudies,atleastonestaroutofthreeismultiple.Itwouldbenicetoseebybrute­force calculationshowacontractionofanirregularlyshapedmassofgasdevelops.Beyondallthisistheproblemoftheformationanddevelopmentofgalaxies—thatis,the assembliesofbillionsofstars.Onthis,too,astrophysicistshaveaccomplishedmuchwiththeaidofcomputers. Whilesuchastrophysicalcalculationsweregoingon,Ibeganinanamateurishwaytoworkonsomequestionsofbiology.Afterreadingaboutthenewdiscoveriesin molecu­

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larbiologywhichwerecomingfast,Ibecamecuriousaboutaconceptualrolewhichmathematicalideascouldplayinbiology.IfImayparaphraseoneofPresident Kennedy'sfamousstatements,Iwasinterestedin"notwhatmathematicscandoforbiologybutwhatbiologycandoformathematics."Ibelievethatnewmathematical schemata,newsystemsofaxioms,certainlynewsystemsofmathematicalstructureswillbesuggestedbythestudyofthelivingworld.Itscombinatorialarrangements mayleadusinthefuturetoalogicandmathematicsofadifferentnaturefromwhatweknownow.Thereaderisreferredtooneofmypapersonmathematical biology.Tootechnicaltobeincludedhere,itislistedinthebibliographyattheendofthisvolume. MyinterestinbiologytookamoretangibleformwhenIengagedindiscussionswithJamesTuck,andwetalkedtothebiologistsinthelaboratory.LosAlamoshad alwayshadadivisionforthestudiesofbiologicaleffectsofradiation.Radioactivedamagewas,ofcourse,oneofthefirstthingstoworryaboutfromthebeginningof thenuclearage.WithTuck,andGordonGouldandDonaldPetersonfromtheHealthDivision,weorganizedaseminardevotedtocurrentproblemsofcellularbiology andthenewresultsinmolecularbiology.Ireallylearnedalotabouttheelementaryfactsofbiologythere,theroleofcells,theirstructure,andsoon.Theseminars, whichhadabouttwentyparticipants,havehadimportantconsequences,althoughtheylastedonlytwoyears.Twooftheparticipants,LosAlamosphysicistsWalter GoadandGeorgeBell,bothextremelybrilliantandtalented,andamongthebestyoungbrainsinthesefieldsinthecountry,aredoingalotofbiologyresearchnow. Goadisworkinginthefieldofbiologicalmathematics,whileBellhassomenewideasonimmunology.TedPuckfromColoradovisitedtheseminarandgavesome lectures. ImetPuckshortlyafterthewarandfoundhimtobefullofnewideas,suggestinginterestingexperimentsandmethodsforthestudyofthebehaviorofcellsand problems

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inmolecularbiologyingeneral.IthinkitwasTedPuck'sgroupwhichfirstsucceededinkeepingmammaliancellsaliveandevenmultiplyinginvitro.Ialwayslook forwardtodiscussionswithhim;itwashewhoarrangedformetogiveseminarsforthefacultyandyoungresearchersinthebiophysicsdepartmentandeven succeededinhavingmeappointedamemberoftheprofessorialstaffattheUniversityofColorado'sMedicalSchool.Itoldhimthat,beingabeginnerandalaymanin thisfield,Imightbearrestedforimpersonatingadoctor. Almosteverymonththerearefascinatingnewfactsdiscoveredinbiology.ItisnowwidelyrecognizedthatthediscoveriesofCrickandWatsonhaveopenedupa newerainthepsychologicalattitudesinbiologyaswell.YearsagoatHarvard,whenItalkedtobiologistsandtriedtoaskaboutorproposeevenamildlygeneral statement,therewouldalwaysbetheretort:"Itisn'tsobecausethereisanexceptioninsuchandsuchaninsect"or"suchandsuchafishisdifferent."Therewasa generaldistrustoratleastahesitationtoformulateanythingofevenaslightlygeneralnature.Thisattitudehasdrasticallychangedsincethediscoveryoftheroleof DNAandthemechanismofreplicationofthecellandofthecodewhichseemsouniversal. DuringalltheseyearsIdidnotlivecontinuouslyinLosAlamos.IspentperiodsoftimeasavisitingprofessoratHarvard,MIT,theUniversityofCaliforniainLaJolla, theUniversityofColorado,plusinnumerablevisitstovariousuniversities,scientificmeetings,andgovernmentorindustriallaboratories,whereIgavelecturesand consultations.Theselatterwerecalledbusinesstrips.IfoneaddsouralmostyearlyvacationsinEuropesince1950(mainlyinFrancewhereFrançoisestillhasrelatives andIhavemanyscientificfriends),itseemstomethatabouttwenty­fivepercentofmytimewasspentawayfromLosAlamos. ItwasinthoseperiodsthatmyfriendshipwithVictorWeisskopfdeveloped.IhadmethiminLosAlamosduringthewarwhenhewasBethe'salternateasleaderof thetheo­

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reticaldivision.Heleftatthewar'sendtobecomeprofessoratMITandourrelationshipdeepenedduringmyvisitstoCambridge,HarvardandMIT. Viki,asheisuniversallycalled,isatheoreticalphysicist.Hemadeanameforhimselfasayoungmanwithhisimportantworkonproblemsofradiationinquantum theory.HewasforatimeassistanttoPauliandalsoworkedinCopenhagenatthefamousNielsBohrInstitute.VikiwasborninVienna,afactItakenoteofbecause heexhibitsthebestsideoftheViennesetemperament.Thisiscontainedinthefollowingsaying:Inpost­World­War­IBerlinpeopleusedtosay,"Thesituationis desperatebutnothopeless";inViennatheysaid,"Thesituationishopelessbutnotserious."Thiscertaininsouciancecombinedwiththehighestintelligencehasenabled Vikitonavigatenotonlythroughtheusualdifficultiesofadministrativeandacademicaffairs—hehasbeen,amongotherthings,director­generalofCERN(the EuropeanCenterforNuclearResearch)nearGenevaandchairmanofthelargephysicsdepartmentatMIT—butalsointhemoreabstractrealmoftheintellectualand scientificdifficultiesoftheoreticalphysics.Iwouldsayhisintellectualstabilityisbasedonarealknowledgeofandfeelingforthespiritofthehistoryofphysics.Thishe hasachievedthroughperspectiveandcomprehension,siftingandevaluationofthequicklychangingsceneinthephysicaltheorieswhichconcerntheveryfoundations ofthisscience.Ishouldaddhereforthebenefitofthereaderwhoisnotaprofessionalphysicistthatthelastthirtyyearsorsohavebeenaperiodofkaleidoscopically changingexplanationsoftheincreasinglystrangeworldofelementaryparticlesandoffieldsofforce.Anumberofextremelytalentedtheoristsviewitheachotherin learnedandcleverattemptstoexplainandordertheconstantflowofexperimentalresultswhich,orsoitseemstome,almostperverselycastdoubtsaboutthejust completedtheoreticalformulations.Throughallthisturmoilintheoverlymathematicaltheoreticalphysicsresearch,constantgoodprogresshasbeenmade,butittakes a

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personlikeViki(andreallytherearenomoresuchthanonecancountonthefingersofonehand)tostabilizethisflowandextractthegistofthenewelaborationsof theideasofquantumtheoryandtobeabletoexplainanddescribeitbothtothephysiciststhemselvesandtothemoregeneralpublic. Hissemipopularbooksonphysicsareuniquelyinterestingandsuccessfulinpresentinghisphilosophyandthehumansideofthestory.Hewasalwaysandstillis immenselyconcernedwiththeproblemsofmanandworldaffairs.Asapersonheisaffableandkind,getsalongwitheverybody,andhelovestotellstories,and sometimesourexchangesofJewishjokescanlastanhour. DuringhistenureatCERNwherehestillvisitseverysummerandconsults,theWeisskopfsbuiltamodestsummerhouseontheFrenchsideoftheborderinasmall JuravillageoverlookingtheLakeofGeneva.Itistwentyminutes'drivefromCERNandtheyspendmostofthesummerthere.OnourownEuropeantripsofthelast severalyearswehavealmostalwaysincludedabriefstaywiththeWeisskopfsintheirhouseinVesancy.Thisvillageisjustafewkilometersawayfromanotherone knownasFerney­VoltairebecauseVoltairelivedthereformanyyears.InlikemannerIhavedubbedVesancy,Vesancy­Weisskopf.Vikilikesthatanditfitsforhe hasbecomequiteapersonageinthevillage.HeisknownasMonsieurleDirecteur,andthefarmerstiptheirhatswhentheyseehistall,lankysilhouettewalking carefullyacrosstheirfields. In1960mybook,UnsolvedProblemsofMathematics,waspublished.ManyyearsagoFrançoiseaskedSteinhauswhatitwasthatmademewhatpeopleseemed toconsiderafairlygoodmathematician.Accordingtoher,Steinhausreplied:''C'estl'hommedumondequiposelemieuxlesproblèmes."Apparentlymyreputation, suchasitis,isfoundedonmyabilitytoposeproblemsandtoasktherightkindofquestions.Thisbookpresentsmyownunsolvedproblems.AsayoungmanIliked themottoinfrontof

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GeorgeCantor'sthesiswhichisaLatinquotation:"Inremathematicaarsproponendiquaestionemplurisfaciendaestquamsolvendi." Shortlyafter1960thebookwastranslatedintoRussian.ThereisnocopyrightagreementbetweenRussiaandtheWest,andtheRussianspaynoroyalties,butsome WesternauthorsdiscoveredwhentheywereintheSovietUnionthattheycouldobtainsomepaymentforthetranslationsoftheirwork.HansBetheandBob Richtmyersuccessfullyreceivedcompensation.SowhenIattendedanInternationalMathematicsCongressinMoscowin1966IrememberthatIcouldtrytoo.The RussianlanguagebeingclosetoPolish,IwenttothepublishinghousetotalkaboutthismatterinmyimitationofRussian.Atthepublishinghouse,whichlookedthe sameaseverywhereelse—girlstypingandmassesoffilesandpapers—anelderlygentlemanseemedtounderstandmyrequestandaskedmehowIknewtocome andseethem.Igavehimthenamesofmyfriends.Hewenttoabackroom,thenreturned.ThereadershouldknowthatRussiansdonotpronounceHasinEnglish. TheysayG.ForinstanceHitlerispronouncedGitler,HamletGamlet,HilbertGilbert.ThegentlemansaidtomeinRussianwithanengagingsmile:"Comeback tomorrowpleasewithyourpassport,andwewillgiveyou"—Iseemedtohear—"yourgonorrhea."Ofcoursehehadreallysaid"gonorar''for"honorar"(honorarium orroyalties).Iwantedtosay,"No,thankyou,"butIunderstoodwhathemeant.ThenextdaywhenIreturnedhehandedmeanenvelopewhichcontainedthree hundredrublesincash.OneisnotallowedtoexportrublesfromRussia,soafterIhadboughtsomesouvenirs,amber,furhats,booksandthelike,Istillhadone hundredrublesleft.IhadtoputtheminapostalsavingsaccountwhichinRussiapaysoneortwopercentinterest.ThismakesmeaSovietUnioncapitalist. Sometimeintheearly1960sImetGian­CarloRota,amathematicianwhoisalmostaquarter­centuryyoungerthanIanddefinitelyrepresentativeofthenext generation.Or

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maybeevenseveralgenerationslater,foracademicallyinmathematicsthegenerationgapmayexistalreadybetweenlecturersandtheirstudentswherechronologically thedifferenceisonlyafewyears.Ourrelationshipisnotbuiltonouragedifference.Rotaclaimsthatheisgreatlyinfluencedbyme.SoIcoinedtheexpression "influencerandinfluencee."Rotaisoneofmybestinfluencees.Banach,forexample,Iconsiderasaninfluencer. FromthestartIwasimpressedbyRota'sfeelingforseveraldifferentmathematicalfieldsandhisopinionsinmanyareasofresearchwhereheexhibitsbotherudition andcommonsense.Itisincreasinglyrarenow—infact,ithasbeenforthelasttwentyyearsormoreinthiseraofincreasingspecialization—tofindapersonwith knowledgeofthehistoricallinesofmathematicaldevelopment. Rotaimpressedmebyhisknowledgeofsomehalf­forgottenfields,theworkofSylvester,Cayleyandothersonclassicalinvarianttheory,andbythewayhemanaged toconnecttheworkofItaliangeometerstoGrassmaniangeometryandmodernizemuchofthisresearchwhichdatestothelastcentury.Hismainfieldofworkwasin combinatorialanalysis,whereagainhemanagedtoupdatesomeclassicalideasandadaptthemtogeometry. IsuggestedthatRotabeinvitedtoLosAlamosforavisitasaconsultant.Hehasbeendoingthisperiodicallyeversinceandprovedveryusefulinseveralways, includingnumericalanalysiswhichisimportantinmanyofthelargecomputationalproblemsworkedontheelectroniccomputers. Rota'spersonalityiscompatiblewithmine.Hisgeneraleducation,activeinterestinphilosophy(heisanexpertontheworkofEdmundHusserlandMartinHeidegger), and,aboveall,hisknowledgeofclassicalLatinandancienthistory,havemadehimfillthegapleftbythelossofvonNeumann.Indeedweoftenvieinquotingfrom Horace,Ovid,andotherauthorsinagood­humoreddisplayofboastingerudition.Rotaisalsoatruebonvivant,exceedinglyfondof

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goodwinesandfoods,especiallythosefromItaly.Heisincrediblyadeptinthepreparationofagreatvarietyofpastadishes.Italianborn,hewasbroughttoSouth AmericarightafterWorldWarII,andattheageofeighteenhecametotheUnitedStates.Hiscollegeeducationtookplacehere,buthehasretainedmanyEuropean mannerismsindress,tastesandhabits.HeisaPrincetongraduateandnowaprofessoratMIT.

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Chapter14 ProfessorAgain 1967—1972 DuringtheLosAlamosyearsIfrequentlytooktimeofftoreturntoacademiclife,andaround1965IstartedvisitingtheUniversityofColoradoonamoreregular basis,soitwasnotadiscontinuouschangewhenin1967IdecidedtoretirefromLosAlamosandacceptaprofessorshipinBoulder.NorwasIgoingtoastrange, newplace;onthecontrary,IwasjoiningseveralofmygoodoldfriendswhohadalsoselectedtheColoradoRockiesasaplacetolive,DavidHawkins,Bob RichtmyerandGeorgeGamow.HawkinshadbeenaprofessorofphilosophyinBouldersincehehadleftLosAlamosafterthewar;Richtmyer,thepost­warleader ofthetheoreticaldivisionbeforeCarsonMark,hadgivenuptheCourantInstituteinNewYorkforthecleanerairofBoulder;Gamowhadbecomeaprofessorinthe physicsdepartmentseveralyearsearlier.TheUniversityofColoradowasflourishingandexpanding,especiallyinthesciences,andthemathematics

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departmentexperiencedanexplosivegrowthinsizeandinquality.Besides,BoulderwassufficientlyclosetoLosAlamos,aneasyday'sdrivethroughspectacular scenery,soIcouldcontinueasaconsultantandvisitfrequently.Thefocusofmyinvolvement,however,shiftedfromLosAlamostoBoulder. InBoulderIsawagreatdealofGamowuntilhisdeathin1968.Hishealthhadbeenfailingforafewyears,hisliverhadweakenedundertheassaultsofalifetimeof carefreedrinking.Hewasquiteawareofthisandsaidtomeonsomeoccasion:"Finallymyliverispresentingmewiththebill."Thisdidnotpreventhimfromworking andwritingtilltheveryend.AthisRussianfuneral,whenhelayinanopencasket,IrealizedthathewasonlytheseconddeadmanIhadeverseeninmylife.ThoughI wasnotconsciousoftheshockthisgaveme,Ihadtoholdontotherailwhenwestoodupforthechantssothatmykneeswouldnotbuckleunderme. Gamow'sautobiography,MyWorldLine,waspublishedposthumouslyfromfragmentsofhisunfinishedmanuscript. Byanincrediblecoincidence,GamowandEdwardCondon,whohaddiscoveredsimultaneouslyandindependentlytheexplanationofradioactivity(oneinRussia,the otherinthiscountry),cametospendthelasttenyearsoftheirliveswithinahundredyardsofeachotherinBoulder.TheyhadbecomefriendseventhoughCondon oftenfeltthatneitherhenorhiscollaboratorGurneyhadreceivedtheirdueshareofcreditforthediscovery. Condonwasamarvelousperson.FormehetypifiedthebestinthenativeAmericancharacter,earthy,super­honest,solid,simple,andatthesametimevery perspicacious.Hispoliticalviewsoftencoincidedwithmine.HedidnotlikeNixon,whohadhoundedhimontheUn­AmericanActivitiesCommitteetothepointthat heresignedfromthedirectorshipoftheBureauofStandards.HejoinedthephysicsdepartmentinBoulderafterdevelopinghearttrouble;theyearbeforehisdeathin 1973,hehadbeengivenanartificial

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heartvalve,whichrenderedhislastmonthsmoreactiveandcomfortable. Intherelativegreaterfreedomofuniversitylife,longervacations,nofixedscheduleexceptforsometeaching,Iwasreturningtoamoreacademictypeofscienceina milieuofmathematiciansandphysicists.Themathematicsdepartmentwasacquiringexcellentresearchersinthefoundationsofmathematics,settheory,logic,and numbertheory.WolfgangSchmidt,anAustrianbybirth,wasoneofthem,powerfulandoriginalinthelatter.Anotherisayounger,brilliantPole,JanMycielski,a studentofSteinhaus,whomIinvitedtoacceptaprofessorshipwhenIwaschairmanofthedepartment.Wehavesincecollaboratedinproblemsofgametheory, combinatorics,settheoryand—duringthelastseveralyears—onmathematicalschemataconnectedwiththestudyofthenervoussystem.Mycielski,withRotaanda LosAlamosmathematician,WilliamBeyer,gatheredandeditedthefirstvolumeofmycollectedworks,whichhasbeenpublishedbytheMITPressunderthetitle Sets,Numbers,Universes.TheBouldermathematicsdepartmentalsohasanumberofyoungpeoplestronginanalysisandtopology. In1967themathematicianMarkKacandIwereinvitedbytheeditorsoftheEncyclopaediaBritannicatowritealongarticlewhichwastobepartofaseriesof specialappendicestoaneweditionoftheBritannica.SincethenithasappearedseparatelyunderthetitleMathematicsandLogic.Itreceivedveryfavorable reviewsandhasbeentranslatedintoFrench,Spanish,Russian,Czech,andJapanese.Itwasratherdifficultforustofindtherightlevelofpresentation.Designednot somuchforthebroadpublicbutratherforscientistsinotherfields,wetriedtomakeitasemi­popularpresentationofmodernideasandperspectivesofthegreat conceptsofmathematics. Asthereadermayhavenoticed,muchofmyworkseemstohavebeendoneincollaborationwithothers(justasthis

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bookwasassembledincollaborationwithFrançoise).Oneofthereasonsforthisismyleaningonconversationasastimulanttothinking;theotherismywell­known impatiencewithdetailandacertaindistasteforreadingwhatIhavewritten.WhenIseeoneofmypapersinprint,Ihaveachildishcomplex,atinynaggingdoubtthat itmightbewrongorthatitmaynotcontainanythinginteresting,andIdiscarditafteraquickglance. MarkKachadalsostudiedinLwów,butsincehewasseveralyearsyoungerthanI(andIhadleftwhenonlytwenty­sixmyself),Iknewhimthenonlyslightly.Hetold methatasayoungstudenthehadbeenpresentatmydoctorateceremonyandhadbeenimpressedbyit.Headdedthatthesefirstimpressionsusuallystay,andthat hestillconsidersme"averyseniorandadvancedperson,"eventhoughtheratioofouragesisnowveryclosetoone.HecametoAmericatwoorthreeyearsafterI did.IrememberedhiminPolandasveryslimandslight,butherehebecameratherrotund.Iaskedhim,acoupleofyearsafterhisarrival,howithadhappened.With hischaracteristicgoodhumorhereplied:"Prosperity!"Hisreadywitandalmostconstantjovialitymakehimextremelycongenial. AfterthewarhevisitedLosAlamos,andwedevelopedourscientificcollaborationandfriendship.AfteranumberofyearsasaprofessoratCornellhebecamea professorofmathematicsatTheRockefellerInstituteinNewYork(nowTheRockefellerUniversity.)HeandthephysicistGeorgeUhlenbeckhaveestablished mathematicsandphysicsgroupsatthisInstitute,wherebiologicalstudiesweretheprincipalandalmostexclusivesubjectbefore. Markisoneoftheveryfewmathematicianswhopossessatremendoussenseofwhattherealapplicationsofpuremathematicsareandcanbe;inthisrespectheis comparabletovonNeumann.HewasoneofSteinhaus'sbeststudents.AsanundergraduatehecollaboratedwithhimonapplicationsofFourierseriesandtransform techniquesto

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probabilitytheory.Theypublishedseveraljointpapersontheideasof"independentfunctions."AlongwithAntoniZygmundheisagreatexponentandtruemasterin thisfield.HisworkintheUnitedStatesisprolific.Itincludesinterestingresultsonprobabilitymethodsinnumbertheory.Inaway,Kac,withhissuperiorcommon sense,asamathematicianiscomparabletoWeisskopfandGamowasphysicistsintheirabilitytoselecttopicsofscientificresearchwhichlieattheheartofthematter andareatthesametimeofconceptualsimplicity.Inaddition—andthisisperhapsrelated—theyhavetheabilitytopresenttoawiderscientificaudiencethemost recentandmodernresultsandtechniquesinanunderstandableandoftenveryexcitingmanner.Kacisawonderfullecturer,clear,intelligent,fullofsenseand avoidanceoftrivia. AmongthemathematiciansofmygenerationwhoinfluencedmethemostinmyyouthwereMazurandBorsuk.MazurIhavedescribedearlier.AsforBorsuk,he representedformetheessenceofgeometricalintuitionandtrulymeaningfultopology.Igleanedfromhim,withoutbeingabletopracticeitmyself,theworkingsofn­ dimensionalimagination.TodayBorsukiscontinuinghiscreativeworkinWarsaw.Hisrecenttheoryofthe"shape"intopologyshowsincreasingpowerand applications.Hisgeneralinterestsandmathematicaloutlookareveryclosetomine,andouroldfriendshipwasrenewedafterthewarduringhisvisitstotheUnited StatesandmybrieftriptoPolandin1973whenIsawhiminhiscountryhousenearWarsaw. Onecouldgoonadinfinitumrecollectingfrommemory,reflectingandwritingdown.Ifthereaderisstillwithme,hemayhavederivedfromtheprecedingasortof existentialist(thewordisinvogue)pictureofmylife,thesetimes,andthemanyscientistsIhaveknown.BywayofaconclusiontothischapterIwilladdaself­portrait whichIsenttoFrançoisebeforeweweremarried.IamtranslatingfromtheFrenchwhichaccountsinpartforitsawkwardness.

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"Self­portraitofMr.S.U. "Hisexpressionisusuallyironicandquizzical.Intruthheisverymuchaffectedbyallthatisridiculous.Perhapshehassometalenttorecognizeandfeelitatonce,soit isnotsurprisingthatthisisreflectedinhisfacialexpression. "Hisconversationisveryuneven,sometimesserious,sometimesgay,butnevertiringorpedantic.Heonlytriestoamuseanddistractthepeoplehelikes.Withthe exceptionoftheexactsciences,thereisnothingwhichappearssocertainorobvioustohimthathewouldnotallowfordifferingopinions:onalmostanysubjectone cansayalmostanything. "Hebroughttothestudyofmathematicsacertaintalentandfacilitywhichallowedhimtomakeanameforhimselfatanearlyage.Dedicatedtoworkandsolitudeuntil hewastwenty­five,hebecamemoreworldlyratherlate.Neverthelessheisneverrudebecauseheisneithercoarsenorhard.Ifhesometimesoffendsitisthrough inattentionorignorance.Inspeechheisneithergallantnorgraceful.Whenhesayskindthingsitisbecausehemeansthem.Thereforetheessenceofhischaracterisa franknessandtruthfulnesswhicharesometimesalittlestrongbutneverreallyshocking. "Impatientandcholerictothepointofviolence,everythingthatcontradictsorwoundshimaffectshiminanuncontrollableway,butthisusuallydisappearswhenhehas ventedhisfeelings. "Heiseasytoinfluenceorgovernprovidedheisunawarethatthisisintended. "Somepeoplethinkthatheismaliciousbecausehemakesmercilessfunofpretentiousbores.Histemperamentisnaturallysensitiveandrendershimsubjecttodelicate moods.Thismakeshimatoncegayandmelancholy. "Mr.U.behavesaccordingtothisgeneralrule:hesaysalotoffoolishthings,seldomwritesthemandneverdoesany.

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WhenFrançoisereadthisdescriptionshefeltitagreedwellwithwhatshethenknewofmebutwasverysurprisedatthequalityofmyFrench,untilshecametothe lastparagraph: "AndnowIshallchangefrommytextwhichIcameuponbychanceyesterday.Theaboveareverbatimextractsfromaletterofd'AlemberttoMademoisellede Lespinassewrittensometwohundredyearsago!"(D'AlembertwasafamousFrenchmathematicianandencyclopedistoftheeighteenthcentury).Francoisewasvery amused. SomethirtyyearshaveelapsedsinceIcopiedthislittletext.IwillnowaddasafinishingtouchthatIdon'tthinkIhavechangedmuch,butthatthereisonetraitwhich d'AlembertdidnotmentionthatIpossess—allthismerelysiparvamagniscompararelicet—itisacertainimpatience.Ihavebeenafflictedwiththisallmylife.It maybeincreasingwithadvancingyears.(IfEinsteinorCantorcametolectureheretodayIwouldhavethesplitreactionofaschoolboy—wantingtolearnontheone handandtoskipclassontheother.)WhileIstillfeelquitehappygivinglectures,talks,ordiscussionsIambecominglessandlessabletositthroughhoursofsuch givenbyothers.Iam,Itoldsomecolleagues,"likeanoldboxerwhocanstilldishitoutbutcan'ttakeitanymore."Thisamusedthemnoend.

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Chapter15 RandomReflectionsonMathematicsandScience Thischapterwillbesomewhatdifferentincontentfromtheprecedingaccountofmy''adventures"andofscientistsIhaveknown. HereIhavetriedtogather,review,andsometimesamplifysomeofthegeneralideasIhavetoucheduponsolightlythroughoutthebook.Ihopethatintheir randomnessthesereflectionswillgivethereaderanaddedglimpseintothemanifoldaspectsofscienceandespeciallytherelationofmathematicstoothersciences.It ismerelyaboutthe"gistofthegist."Forgreaterdetail,Icanonlyreferthereadertosomeofmymoregeneralscientificpublications. Whatexactlyismathematics?Manyhavetriedbutnobodyhasreallysucceededindefiningmathematics;itis

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alwayssomethingelse.Roughlyspeaking,peopleknowthatitdealswithnumbersandfigures,withpatterns,relations,operations,andthatitsformalprocedures involvingaxioms,proofs,lemmas,theoremshavenotchangedsincethetimeofArchimedes.Theyalsoknowthatitpurportstoformthefoundationsofallrational thought. Somecouldsayitistheexternalworldwhichhasmoldedourthinking—thatis,theoperationofthehumanbrain—intowhatisnowcalledlogic.Others—philosophers andscientistsalike—saythatourlogicalthought(thinkingprocess?)isacreationoftheinternalworkingsofthemindastheydevelopedthroughevolution "independently"oftheactionoftheoutsideworld.Obviously,mathematicsissomeofboth.Itseemstobealanguagebothforthedescriptionoftheexternalworld, andpossiblyevenmoresofortheanalysisofourselves.Initsevolutionfromamoreprimitivenervoussystem,thebrain,asanorganwithtenormorebillionneurons andmanymoreconnectionsbetweenthemmusthavechangedandgrownasaresultofmanyaccidents. Theveryexistenceofmathematicsisduetothefactthatthereexiststatementsortheorems,whichareverysimpletostatebutwhoseproofsdemandpagesof explanations.Nobodyknowswhythisshouldbeso.Thesimplicityofmanyofthesestatementshasbothaestheticvalueandphilosophicalinterest. Theaestheticsideofmathematicshasbeenofoverwhelmingimportancethroughoutitsgrowth.Itisnotsomuchwhetheratheoremisusefulthatmatters,buthow elegantitis.Fewnon­mathematicians,evenamongotherscientists,canfullyappreciatetheaestheticvalueofmathematics,butforthepractitionersitisundeniable. Onecan,however,lookconverselyatwhatmightbecalledthehomelysideofmathematics.Thishomelinesshastodowithhavingtobepunctilious,ofhavingtomake sureofeverystep.Inmathematicsonecannotstopatdrawingwithabig,widebrush;allthedetailshavetobefilledinatsometime.

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"Mathematicsisalanguageinwhichonecannotexpressunpreciseornebulousthoughts,"saidPoincaré,Ibelieveinaspeechonworldsciencewhichhegaveatthe St.Louisfairmanyyearsago.AndhegaveasanexampleoftheinfluenceoflanguageonthoughtadescriptionofhowdifferentlyhefeltusingEnglishinsteadof French. Itendtoagreewithhim.ItisatruismtosaythatthereisaclaritytoFrenchwhichisnotthereinothertongues,andIsupposethismakesadifferenceinthe mathematicalandscientificliterature.Thoughtsaresteeredindifferentways.InFrenchgeneralizationscometomymindandstimulatemetowardconcisenessand simplification.InEnglishoneseesthepracticalsense;Germantendstomakeonegoforadepthwhichisnotalwaysthere. InPolishandRussian,thelanguagelendsitselftoasortofbrewing,adevelopmentofthoughtliketeagrowingstrongerandstronger.Slaviclanguagestendtobe pensive,soulful,expansive,morepsychologicalthanphilosophical,butnotnebulousorcarriedbywordsasmuchasGerman,wherewordsandsyllablesconcatenate. Theyconcatenatethoughtswhichsometimesdonotgoverywelltogether.Latinissomethingelseagain.Itisorderly;clarityisalwaysthere;wordsareseparated;they donotgluetogetherasinGerman;itislikewell­cookedricecomparedtoovercooked. Generallyspeaking,myownimpressionsoflanguagesarethefollowing:WhenIspeakGermaneverythingIsayseemsoverstated,inEnglishonthecontraryitfeels likeanunderstatement.OnlyinFrenchdoesitseemjustright,andinPolish,too,sinceitismynativelanguageandfeelssonatural. SomeFrenchmathematiciansusedtomanagetowriteinamorefluentstylewithoutstatingtoomanydefinitetheorems.Thiswasmoreagreeablethanthepresentstyle oftheresearchpapersorbookswhichhavesomuchsymbolismandformulaeoneverypage.IamturnedoffwhenIseeonlyformulasandsymbols,andlittletext.It istoolaborious

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formetolookatsuchpagesnotknowingwhattoconcentrateon.Iwonderhowmanyothermathematiciansreallyreadthemindetailandenjoythem. Theredoexist,though,important,laboriousandineleganttheorems.Forexamplesomeoftheworkconnectedwithpartialdifferentialequationstendstobeless "beautiful"informandstyle,butitmayhave"depth,"andmaybepregnantwithconsequencesforinterpretationsinphysics. Howdoesonearriveatavaluejudgmentnowadays? Mathematicians,whosejobinasenseistoanalyzethemotivationandoriginoftheirwork,foolthemselvesandmayberemisswhentheythinktheirmainbusinessisto provetheoremswithoutatleastindicatingwhytheymaybeimportant.Ifleftentirelytoaestheticcriteria,doesn'titcompoundthemystery? Ibelievethatinthedecadestocometherewillbemoreunderstanding,evenonaformallevel,ofthedegreeofbeauty,thoughbythattimethecriteriamayhaveshifted andtherewillbeasuperbeautyinunanalyzablehigherlevels.Sofarwhenanyonehastriedtoanalyzetheaestheticcriteriaofmathematicstooprecisely,whateverwas proposedhasseemedtoonarrow.Ithastoappealtoconnectionswithothertheoriesoftheexternalworldortothehistoryofthedevelopmentofthehumanbrain,or elseitispurelyaestheticandverysubjectiveinthesensethatmusicis.Ibelievethateventhequalityofmusicwillbeanalyzable—toanextentonly,ofcourse—atleast byformalcriteria,bymathematizingtheideaofanalogy. Someoftheoldproblems,unsolvedformanyyears,arebeingsettled.Somearesolvedwithabang,andotherswithawhimper,sotospeak.Thisappliestoproblems seeminglyequallyimportantandaprioriinteresting,butsome,evenfamousclassicalones,aresolvedinsuchaspecificwaythatthereisnothingmoretobeaskedor said.Someothers,lessfamous,immediatelyuponsolutionbecomesourcesofcuriosityandactivity.Theyseemtoopennewvistas. Asforpublications,mathematiciansnowadaysarealmost

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forcedtoconcealthewaytheyobtaintheirresults.EvaristeGalois,theyoungFrenchgeniuswhodiedattheageoftwenty­one,inhislastletterwrittenbeforehisfatal duel,stressedhowtherealprocessofdiscoveryisdifferentfromwhatfinallyappearsinprintastheprocessofproof.Itisimportanttorepeatthisagainandagain. Onthewholeandinthelargelinestheredoesseemtoexistaconsensusamongworkingmathematiciansaboutthevalueofindividualachievementsandthevalueof newtheories.Theremustthereforebesomethingobjectiveifnotyetdefinedaboutthefeelingofbeautywhichmathematicsoffers,dependentsometimesalsoonhow usefulitturnsouttobeinotherbranchesofitselforofothersciences.Whymathematicsisreallysousefulinthedescriptionofthephysicalworld,formeatleast remainsphilosophicallyamystery.EugeneWigneroncewroteafascinatingarticleonthis"implausible"usefulnessofmathematicsandtitledit"TheUnreasonable EffectivenessofMathematics." Itis,ofcourse,oneveryconcisewayofformalizingallrationalthought. Italsohasmanifestly,inelementary,secondaryandadvancedschoolsthevalueoftrainingourbrain,sincepractice,justasinanyothergame,sharpenstheorgan.I cannotsaywhetheramathematician'sbrainistodaysharperthanitwasinthetimeoftheGreeks;neverthelessonthelongerscaleofevolutionitmustbeso.Ido believemathematicsmayhaveagreatgeneticrole,itmaybeoneofthefewmeansofperfectingthehumanbrain.Iftrue,nothingcouldthenbemoreimportantfor humanity,whethertoarriveatsomenewdestinyasagrouporforindividuals.Mathematicsmaybeawayofdevelopingphysically,thatisanatomically,new connectionsinthebrain.Ithasasharpeningvalueeventhoughtheenormousproliferationofmaterialshowsatendencytobeatthingstodeath. Yeteveryformalism,everyalgorithm,hasacertainmagicinit.TheJewishTalmud,oreventheKabbalah,containsmaterialwhichdoesnotappearparticularly enlighteningintel­

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lectually,beingavastcollectionofgrammaticalorculinaryrecipes,someperhapspoetic,othersmystical,allratherarbitrary.Overcenturiesthousandsofmindshave poredover,memorized,dissected,andclassifiedtheseworks.Insodoingpeoplemayhavesharpenedtheirmemoriesanddeductivepractice.Asonesharpensa knifeonawhetstone,thebraincanbesharpenedondullobjectsofthought.Everyformofassiduousthinkinghasitsvalue. Thereexistinmathematicspropositions,suchastheonecalled"Fermat'sGreatTheorem,"which,standingbythemselves,seemspecialandunrelatedtothemain bodyofnumbertheory.Theyareverysimpletostatebuthavedefiedalltheeffortsofthegreatestmindstoprovethem.Suchstatementshavestimulatedyoungminds (myownincluded)tomoregeneralwonderandcuriosity.InthecaseofFermat'sproblem,specialorirrelevantasitisbyitself,ithasstimulatedthroughthelastthree centuriesofmathematics,thecreationofnewlivingobjectsofmathematicalthought,inparticulartheso­calledtheoryofidealsinalgebraicstructures.Thehistoryof mathematicsknowsanumberofsuchcreations. Theinventionofimaginaryandcomplexnumbers(whicharepairsofrealnumberswithaspecialrulefortheiradditionandmultiplication)beyondtheimmediate purposeandtheusetowhichtheywereput,openednewpossibilitiesandledtothediscoveryofmiraculouspropertiesofthecomplexvariables.Theseanalytic functions(theexamplesofwhichare,tomentionthesimplest,

),possessunexpected,simpleandaprioriunforeseenpropertiesderivingfromthefewgeneralruleswhichgovernthem.Theyhaveconvenientalgorithmsandrather deepconnectionswiththepropertiesofgeometricalobjectsandalsowiththemysteriesconcerningtheseeminglysofamiliarnaturalnumbers,theordinaryintegers.It isasifsomeinvisibledifferentuniversegoverningourthoughtbecamedimlyperceptiblethroughit,auniverse

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withsomelaws,andyes,facts,ofwhichwebecomeonlyvaguelyaware. Thefactthatsomeseeminglyveryspecialfunctions,liketheRiemannZetafunction,havesuchdeepconnectionswiththebehaviorofintegers,ofprimenumbers,is hardtoexplainaprioriandindepth.Thisisreallynotwellunderstoodtothisday.Somehowtheseentities,thesespecialanalyticfunctionsdefinedbyinfiniteseries, havebeengeneralizedmorerecentlytospacesotherthantheplaneofallcomplexnumbers,suchastoalgebraicsurfaces.Theseentitiesshowconnectionsbetween seeminglydiversenotions.Theyalsoseemtoshowtheexistence(tomakeametaphorstimulatedbythesubjectitself)ofanothersurfaceofreality,anotherRiemann surfaceofthought(andconnectionsofthought)ofwhichwearenotconsciouslyaware. Someofthepropertiesoftheanalyticfunctionsofthecomplexnumbersturnouttobenotmerelyconvenient,butveryfundamentallytiedtophysicalpropertiesof matter,inthetheoryofhydrodynamics,inthedescriptionofthemotionsofincompressiblefluidssuchaswater,inelectrodynamics,andinthefoundationsofquantum theoryitself. Thecreationofageneralideaofaspace,abstractedtobesure,butnotreallycompletelydictatedoruniquelyindicatedbythephysicalspaceofoursenses,the generalizationtothen­dimensionalspacewherenisgreaterthanthreeandeventoinfinitelymanydimensions,andsousefulatleastasalanguageforthefoundations ofphysicsitself,arethesemarvelsofthepowerofthehumanbrain?Orisitthenatureofthephysicalrealitywhichrevealsittous?Theveryinvention,orisit "discovery,"thattherearedifferentdegreesordifferentkindsofinfinityhashadnotonlyaphilosophical,butbeyondthat,astrikingpsychologicalinfluenceon receptiveminds. Speakingofthefascinationofsurprises,themysteriousattractionofmathematicsand,ofcourse,ofothersciences—

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physicsespecially—itmayberemarkedhowoftenithappensthatinthegameofchessonemayobserveweakplayersorevenrankbeginnersgettingintodeepand fascinatingpositions.Ihaveoftenwatchedamateursornon­talentedbeginners,lookedattheirgameaftersomefifteenmoves,observedthattheirpositionarrivedat perhapsbychance,certainlynotbydesign,wasfullofmarvelouspossibilitiesforbothsides.AndIwonderhowitisthatthegameitselfproducesthesepositionsof greatappealandartwithoutthesesimplefellowsbeingevenawareofit.IdonotknowwhetherananalogousexperienceispossibleinthegameofGo.Icannot myselfjudge,notknowingmuchabouttheintricaciesofthatbeautifulgame,butIwonderwhetheramasterlookingatapositioncantellwhetheritwasarrivedatby chanceorbyalogicallydevelopedcorrectandthoughtfulplay. Inscience,andinmathematicsinparticular,thereseemstobeasimilarmagicalinterestincertainalgorithms.Theyappeartohaveapowertoproducebythemselves, asitwere,solutionstoproblemsorvistasofnewperspectives.Whatseemedtobeatfirstmeretoolsdesignedforspecialpurposescanbringaboutsomeunforeseen andunexpectednewuses. Bytheway,alittlephilosophicalconundrumoccurredtomewhichIdonotknowhowtoresolve:Consideragamelikeasolitaire,oragamebetweentwopersons. Assumethattheplayersmaycheatonceortwiceduringthecourseofthegame.Forinstance,inaCanfieldsolitaire,ifonechangesthepositionofoneortwocards onceandonceonly,thegameisnotdestroyed.Itwouldstillbeaprecisely,completely,mathematicallymeaningful,albeitdifferentgame.Itwouldbecomesimplyabit richer,moregeneral.Butifonetakesamathematicalsystem,asystemofaxiomsandallowstheadditionofoneortwofalsestatements,theresultisimmediate nonsensebecauseonceonehasafalsestatement,onecandeduceanythingonewantsto.Wheredoesthedif­

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ferencelie?Perhapsitliesinthefactthatonlyacertainclassofmotionsisallowedinthegame,whereasinmathematicsonceanincorrectstatementisintroducedone mayimmediatelygetthestatement:zeroequalsone.Theremustthenbeawaytogeneralizethegameofmathematicssothatonecouldmakeafewmistakesand insteadofgettingcompletenonsense,obtainmerelyawidersystem. HawkinsandIhavespeculatedonthefollowingrelatedproblem:avariationonthegameofTwentyQuestions.Someonethinksofanumberbetweenoneandone million(whichisjustlessthan220).Anotherpersonisallowedtoaskuptotwentyquestions,toeachofwhichthefirstpersonissupposedtoansweronlyyesorno. Obviouslythenumbercanbeguessedbyaskingfirst:Isthenumberinthefirsthalfmillion?thenagainreducethereservoirofnumbersinthenextquestionbyone­half, andsoon.Finallythenumberisobtainedinlessthanlog2(1,000,000).Nowsupposeonewereallowedtolieonceortwice,thenhowmanyquestionswouldone needtogettherightanswer?Oneclearlyneedsmorethannquestionsforguessingoneof2nobjectsbecauseonedoesnotknowwhentheliewastold.Thisproblem isnotsolvedingeneral. InmybookonunsolvedproblemsIclaimthatmanymathematicaltheoremscanbepayzised(aGreekwordwhichmeanstoplay).Thatis,thattheycanbeformulated ingame­theoreticallanguage.Forexample,arathergeneralschemaforplayingagamecanbesetupasfollows: SupposeNisagivenintegerandtwoplayersaretobuildtwopermutationsofNletters(n1,n2,...nN).Theyareconstructedbythetwoplayersinturn,asfollows. Firstpermutation,thefirstplayertakesn1,thesecondn2,thefirsttakesn3andsoon.Finallythefirstpermutationisaccomplished.Thentheyplayforthesecond permutation,andifthetwopermutationsgeneratethegroupofallpermutations,thefirstplayerwins,ifnotthesecondwins.Whohasawinningstrategyinthisgame? Thisismerelyasmallexampleof

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how,inanydomainofmathematics—inthiscaseinfinitegrouptheory—onecaninventgamelikeschematawhichleadtopurelymathematicalproblemsandtheorems. Onecanalsoaskquestionsofadifferenttype:Ifthisisdoneatrandomwhatarethechances?Thisisaproblemthatcombinesmeasuretheory,probabilityand combinatorics.Onemayproceedthiswayinmanydomainsofmathematics. Settheoryrevolutionizedmathematicstowardtheendofthenineteenthcentury.WhatstartedthiswasthatGeorgCantorproved(i.e.,discovered)thatthecontinuum isnotcountable.Hedidhavepredecessorsinthesespeculationsonthelogicofinfinity,WeierstrassandBolzano,butthefirstprecisestudyofdegreesofinfinitywas certainlyhis.Thisarosefromhisdiscussionoftrigonometricseries,andveryquicklytransformedtheshapeandflavorofmathematics.Itsspirithasincreasingly pervadedmathematics;recentlyithashadanewandtechnicallyunexpectedreneweddevelopment,notonlyinitsmostabstractform,butalsoinitsimmediate applications.Theformulationsoftopology,ofalgebraicalideasintheirmostgeneralformreceivedimpetusanddirectionfromtheactivitiesofthePolishschool,much ofitfromLwówwheretheinterestscenteredaroundwhatcanbecalledroughlyfunctionalanalysisinageometricalandalgebraicalspirit. Togiveanoversimplifieddescriptionoftheoriginofmuchoftheseactivities:AfterCantorandthemathematiciansoftheFrenchschool,Borel,Lebesgueandothers, thiskindofinvestigationfoundahomeinPoland.InherbookIllustriousImmigrants,LauraFermiexpressesasurprisedadmirationatthelargepercentageofPolish mathematiciansintheUnitedStateswhocontributedsomuchsignificantworktotheflourishingofthisfield.Manycameheretosettleandcontinuesuchwork. SimultaneouslythestudiesofanalysisofHilbertandotherGermanmathematiciansbroughtaboutasimplegeneralmathematicalconstructionofinfinitelydimensional functionalspaces,alsolaterfurtherdevelopedbythePolishschool.Independentlyandatthe

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sametime,theworkofMoore,Veblen,andothersinAmericabroughtaboutameetingofthegeometricalandalgebraicalpointsofview,andaunification—onlyto someextenttobesure—ofmathematicalactivities. Itappearsthatinspiteofincreasingdiversityandevenoverspecialization,thechoiceofmathematicaltopicsofresearchthusfollowsprevalentcurrents,threads,and trendswhichcometogetherfromindependentsources. Somefewindividualswithafewnewdefinitionsareapparentlyabletostartawholeavalancheofworkinspecialfields.Thisispartlyduetofashionandself­ perpetuationbythesheerforceoftheteachers'influence.WhenIfirstcametothiscountryIwasamazedatwhatseemedtomeanexaggeratedconcentrationon topology.NowIfeelthereisperhapstoomuchworkinthedomainofalgebraicgeometry. AsecondepochallandmarkwasGödel'swork,recentlymademorespecificbyPaulCohen'sresults.Gödel,themathematicallogicianattheInstituteforAdvanced StudiesinPrinceton,foundthatanyfinitesystemofaxiomsorevencountablyinfinitesystemsofaxiomsinmathematics,allowsonetoformulatemeaningfulstatements withinthesystemwhichareundecidable—thatistosay,withinthesystemonewillnotbeabletoproveordisprovethetruthofthesestatements.Cohenopenedthe doortoawholeclassofnewaxiomsofinfinities.Thereisnowaplethoraofresultsshowingthatourintuitionofinfinityisnotcomplete.Theyopenupmysteriousareas inourintuitionstodifferentconceptsofinfinity.Thiswill,inturn,contributeindirectlytoachangeinthephilosophyoffoundationsofmathematics,indicatingthat mathematicsisnotafinishedobjectaswasbelieved,basedonfixed,uniquelygivenlaws,butthatitisgeneticallyevolving.Thispointofviewhasnotyetbeen acceptedconsciously,butitpointsawaytoadifferentoutlook.Mathematicsreallythrivesontheinfinite,andwhocantellwhatwillhappentoourattitudestoward thisnotionduringthenextfiftyyears?Certainly,therewillbesomething—ifnotaxiomsinthepresentsenseoftheword,atleastnew

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rulesoragreementsamongmathematiciansabouttheassumptionofnewpostulatesorratherletuscallthemformalizeddesiderata,expressinganabsolutefreedomof thought,freedomofconstruction,givenanundecidableproposition,inpreferencetotrueorfalseassumption.Indeedsomestatementsmaybeundecidably undecidable.Thisshouldhavegreatphilosophicalinterest. Theinterestinthefoundationsofmathematicsistosomeextentalsophilosophical,thougheventuallyitdoespervadeeverything,likesettheory.Buttheword ''foundations"isamisnomer;forthetimebeing,itisjustonemoremathematicalspecialty,fundamentaltobesure. Thegreatatdichotomyintheoriginandintheinspirationofmathematicalthought—stimulatedbytheinfluenceofexternalreality,thephysicaluniverse,ononehand, andbythedevelopingprocessesofphysiology,almostperhapsofthehumanbrain,ontheother—hasinasmallandspecialwayahomomorphicimageinthepresent andnear­futureuseofelectroniccomputers. Eventhemostidealisticpointofviewaboutmathematicsasapurecreationofthehumanmindmustbereconciledwiththefactthatthechoiceofdefinitionsand axiomsofgeometry—infactofmostmathematicalconcepts—istheresultofimpressionsobtainedthroughoursensesfromexternalstimuliandinherentlyfrom observationsandexperimentsinthe"externalworld."Thetheoryofprobability,forexample,cameaboutasadevelopmentofafewquestionsconcerninggamesof chance.Now,computingmachinesconstructedtosolvespecialproblemsofmathematicspromisetoenlargeverygreatlythescopeoftheGedankenExperimente,the idealizationofexperience,andourmoreabstractschemataofthought. Itappearsthatexperimentationonmodelsofgamesplayedbyself­organizinglivingmatterthroughchemicalreactionsinlivingorganismswillleadtonovelabstract mathematicalschemata.Thenewstudyofthemathematicsofgrowingpatterns,andthepossibilityofstudyingexperi­

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mentallyoncomputingmachinesthecourseofcompetitionsorcontestsbetweengeometricalconfigurationsimitatingthefightforsurvival,thesemightgiverisetonew mathematicalsetups.Onecouldagaingivenameslike"payzonomy"tothecombinatoricsofcontestingreactionsand"auxology"toayet­to­be­developedtheoryof growthandorganization,thislatterultimatelyincludingthegrowingtreeofmathematicsitself. Sofaronlytheverysimplestandcrudemathematicalschematahavebeenproposedtomirrorthemathematicalpropertiesofgeometricalgrowth.(Anaccountofmy ownsimple­mindedmodelscanbefoundinarecentbookeditedbyArthurBurke,ATheoryofCellularAutomata,publishedbytheUniversityofIllinoisPress.) AnespeciallyingenioussetofruleswasdevisedbytheEnglishmathematicianJohnConway,anumbertheorist.TheConwayGameofLifeisanexampleofagameor pastimewhich,perhapsmuchliketheearlyproblemsinvolvingdiceandcards,hasledultimatelytothepresentedificeofprobabilitytheory,andmayleadtoavast newtheorydescribingthe"processes"whichAlfredNorthWhiteheadstudiedinhisphilosophy. Theuseofcomputersseemsthusnotmerelyconvenient,butabsolutelyessentialforsuchexperimentswhichinvolvefollowingthegamesorconteststhroughavery greatnumberofmovesorstages.Ibelievethattheexperiencegainedasaresultoffollowingthebehaviorofsuchprocesseswillhaveafundamentalinfluenceon whatevermayultimatelygeneralizeorperhapsevenreplaceinmathematicsourpresentexclusiveimmersionintheformalaxiomaticmethod. Thealready­mentionedrecentresultsofPaulCohenandothers—PeterNovikoff,HaoWang,YuriMatiasevic—ontheindependencefromthetraditionalsystemof axiomsofsomeofthemostfundamentalmathematicalstatements,indicateanewroleforpragmaticapproaches.Workwithautomatawillhelpindicatewhethera problemcanbesolvedbyexistingmeans.

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Toillustratewhatwehaveinmindletusconsiderforexamplea"little"specialprobleminthreedimensions:givenaclosedcurveinspaceandasolidbodyofgiven shape,theproblemistopushthebodythroughthecurve.Therearenoclearmathematicalcriteriatodecidewhetheritcanbedoneornot.Onehastorotate,wiggle, squeeze,and"try,"toseewhetheritcanbedone.Inahighernumberofdimensions,likefive,onecanhaveananalogousproblem.Theideaistosetitupona computerandtryvariouspossiblemotions.Perhaps,afterverymanytries,onewouldacquireafeelingforthefreedomofmaneuveringinthishighdimensionalspace andanewtypeofanalmosttactileintuition.Ofcourse,thisisaspecial,smallandunimportantexample,butIfeelthatonecoulddevelopnewimaginationsbysuitable experimentationwiththesenewtools,electroniccomputersespecially,insettingupandobservingthevariousgrowthprocessesandevolutionarydevelopments. Itseemstometheimpactandroleoftheelectroniccomputerwillsignificantlyaffectpuremathematicsalso,justasithasalreadydonesointhemathematicalsciences, principallyinphysics,astronomyandchemistry. TheseconjecturedexcursionsintoaspectsofthefutureofmathematicstakeusfarfromvonNeumannandhiscontemporaries,andtheirroleintheevolutionofscience aquarter­centuryago.Therateofgrowthintheorganizedactivitiesofthehumanmind,acceleratednodoubtbytheadventofcomputers,seemstoincreaseinaway whichforebodesqualitativechangesinourwayofthinkingandliving.AsNielsBohrsaidinoneofhisamusingremarks:"Itisveryhardtopredict,especiallythe future."ButIthinkmathematicswillgreatlychangeitsaspect.Somethingdrasticmayevolve,anentirelydifferentpointofviewontheaxiomaticmethoditself.Instead ofdetailedworkonspecialtheoremswhichnownumberinthemillions,insteadofthinkingintermsofrulesoperatingwithsymbolsgivenonceandforall,itmaybe thatmathematicswillconsistmoreandmoreofproblems,ordesiderata,orprogramsforworkofageneral

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nature.Nolongerwilltherebeadditionalmultitudesofspecialspaces,definitionsofspecialmanifolds,ofspecialmappingsofthisandthat—thoughafewwillsurvive: "apparentrarinantesingurgitevasto,"nonewcollectionsofindividualtheorems,butinsteadgeneralsketchesoroutlinesoflargertheories,ofvasterenterprises,and theactualworkingoutofproofsoftheoremswillbelefttostudentsoreventomachines.Itmaybecomecomparabletoimpressionisticpaintingincontrasttothe painful,detaileddrawingofearlierdays.Itcouldbeamorelivingandchangingscene,notonlyinthechoiceofdefinitionsbutintheveryrulesofthegame,thisgreat gamewhoserulesuntilnowhavenotchangedsinceantiquity. Iftheruleshavenotchanged,verygreatchangeshavealreadytakenplaceinthescopeofmathematicsinthespaceofmyownlifetime.Inthenineteenthcenturythe applicationsofmathematicswereall­inclusiveinphysics,astronomy,chemistry,inmechanics,engineering,andalltheotherfacetsoftechnology.Morerecently, mathematicsservestoformulatethefoundationsofothersciencesaswell,so­calledmathematicalphysicsisreallythetheoryofallphysics,reachingintoitsmost abstractpartslikequantumtheory,theverystrangefour­dimensionalcontinuumofspace­time.Thesebelongspecificallytothetwentiethcentury.Intheshortspanof sixtytoonehundredyearstheproliferationoftheuseofmathematicalideashasbeenunbelievablyvaried.Itwasaccompaniedby,onecouldsay,anexplosive creationofnewmathematicalobjects,largeandsmall,andatendencyto"beatthingstodeath"withproliferationandhairsplittingstudiesofminutedetailsthatisalmost Talmudic. AtatalkwhichIgaveatacelebrationofthetwenty­fifthanniversaryoftheconstructionofvonNeumann'scomputerinPrincetonafewyearsago,Isuddenlystarted estimatingsilentlyinmymindhowmanytheoremsarepublishedyearlyinmathematicaljournals.(Atheorembeingdefinedasastatementwhichisjustlabeled "theorem,"andispub­

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lishedinarecognizedmathematicaljournal.)Imadeaquickmentalcalculation,amazingmyselfthatIcoulddothiswhiletalkingaboutsomethingentirelydifferentand cametoanumberlikeonehundredthousandtheoremsperyear.QuicklychangingmytopicImentionedthisandtheaudiencegasped.Itmayinterestthereaderthat thenextdaytwooftheyoungermathematiciansintheaudiencecametotellmethatimpressedbythisenormousfiguretheyundertookamoresystematicanddetailed searchintheInstitutelibrary.Bymultiplyingthenumberofjournalsbythenumberofyearlyissues,bythenumberofpapersperissueandtheaveragenumberof theoremsperpaper,theirestimatecametonearertwohundredthousandtheoremsayear.Suchanenormousnumbershouldcertainlygivefoodforthought.Ifone believesthatmathematicsismorethangamesandpuzzles,hereissomethingtoworryabout.Clearlythedangeristhatmathematicsitselfwillsufferthefateofsplitting intodifferentseparatesciences,intomanyindependentdisciplinestenuouslyconnected.Myownhopeisthatthiswillnothappen,forifthenumberoftheoremsis largerthanonecanpossiblysurvey,whocanbetrustedtojudgewhatis"important"?Theproblembecomesoneofrecordkeeping,ofstorageandretrievalofthe resultsobtained.Thisproblemnowbecomesparamount;onecannothavesurvivalofthefittestifthereisnointeraction. Itisactuallyimpossibletokeepabreastofeventhemoreoutstandingandexcitingresults.Howcanonereconcilethiswiththeviewthatmathematicswillsurviveasa singlescience?Justasonecannotknowallthebeautifulwomenorallthebeautifulworksofartandonefinallymarriesonebeautifulperson,onecansaythatin mathematicsonebecomesmarriedtoone'sownlittlefield.Becauseofthis,thejudgmentofvalueinmathematicalresearchisbecomingmoreandmoredifficult,and mostofusarebecomingmainlytechnicians.Thevarietyofobjectsworkedonbyyoungscientistsisgrowingexponentially.Perhapsoneshouldnotcallitapollutionof thought;itispossiblyamir­

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roroftheprodigalityofnaturewhichproducesamillionspeciesofdifferentinsects.Somehowonefeels,though,thatitgoesagainstthegrainofone'sidealsofscience, whichaimstounderstand,abbreviate,summarize,and,inparticular,todevelop,anotationsystemforthephenomenaofthemindandofnature. Itistheunexpectedinthedevelopmentofscience,thewayreallynewideasandconceptsstrikeayoungmind,thatmolditirreversibly.Later,forthematureorolder mind,theunexpectedcausesawonderwhichinducesnewstimulation,evenwhenonehasbecomelessimpressionableorevenjaded.ToquoteEinstein,"Themost beautifulthingwecanexperienceisthemysterious.Itisthesourceofalltrueartandscience." Mathematicscreatesnewobjectsofthought—onecouldcallitameta­reality—byengenderingideaswhichbegintolivetheirownlifeinanindependentdevelopment. Oncebornthesecannotanymorebecontrolledbyasingleperson,onlybyacollectionofbrainswhicharetheperpetuatingsetofmathematicians. Talentorgeniusinmathematicsishardtoquantify.ItendtofeelthatthereisanalmostcontinuouspassagefrommediocritytothehighestlevelsofpeoplelikeGauss, PoincaréandHilbert.Somuchdependsnotonthebrainalone.TherearedefinitelywhatIhavecalled,forwantofabetterword,"hormonalfactors"ortraitsof character:stubbornness,physicalability,willingnesstowork,whatsomecall"passion."Thesedependagreatdealonhabitsmostlyacquiredinchildhoodorearly youthwhenaccidentsofearlyimpressionsplayagreatrole.Undoubtedly,muchofthequalitycalledimaginationorintuitioncomesfromthephysiologicalstructural propertiesofthebrain,whichinturnmaybepartlydevelopedthroughexperiencesleadingtocertainhabitsofthoughtandofthedirectionofthetrainofthought. Thewillingnesstoplungeintotheunknownandtheunfamiliarvarieswithdifferentindividuals.Therearedistinctlydifferenttypesofmathematicians—thosewhoprefer

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toattackexistingproblemsortobuildonwhatisalreadythere,andthosewholiketoimaginenewschemataandnewpossibilities.Thefirstperhapsconstitutea majority,maybemorethaneightypercent.Whenayoungmanwantstoestablishareputationhewillmostlyattackanunsolvedproblemthathasalreadybeenworked on.Inthisway,ifheisluckyorstrongenough,itwillbecomparabletoanathletebeatingarecord,jumpinghigherthananyonebefore.Althoughwhatisoftenof greatervalueistheconceptionofanewidea,ayoungpersonisoftenunwillingtotrythis,notknowingwhetherthenewthoughtwillbeappreciatedevenwhentohim itisimportantandbeautiful. Iamofthetypethatlikestostartnewthingsratherthanimproveorelaborate.Thesimplerand"lower"IcanstartthebetterIlikeit.Idonotrememberusing complicatedtheoremstoprovemorecomplicatedones.(Ofcourse,thisisallrelative,"there'snothingnewunderthesun"—everythingcanbetracedbackto Archimedesorevenearlier.) Ialsobelievethatchangingfieldsofworkduringone'slifeisrejuvenating.Ifonestaystoomuchwiththesamesubfieldsorthesamenarrowclassofproblemsasortof self­poisoningpreventsacquisitionofnewpointsofviewandonemaybecomestale.Unfortunately,thisisnotuncommoninmathematicalcreativity. Withallitsgrandiosevistas,appreciationofbeauty,andvisionofnewrealities,mathematicshasanaddictivepropertywhichislessobviousorhealthy.Itisperhaps akintotheactionofsomechemicaldrugs.Thesmallestpuzzle,immediatelyrecognizableastrivialorrepetitivecanexertsuchanaddictiveinfluence.Onecanget drawninbystartingtosolvesuchpuzzles.IrememberwhentheMathematicalMonthlyoccasionallypublishedproblemssentinbyaFrenchgeometerconcerning banalarrangementsofcircles,linesandtrianglesontheplane."Belanglos,"astheGermanssay,butneverthelessthesefigurescoulddrawyouinonceyoustartedto thinkabouthowtosolvethem,even

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whenrealizingallthetimethatasolutioncouldhardlyleadtomoreexcitingormoregeneraltopics.ThisismuchincontrasttowhatIsaidaboutthehistoryofFermat's theorem,whichledtothecreationofvastnewalgebraicalconcepts.Thedifferenceliesperhapsinthatlittleproblemscanbesolvedwithamoderateeffortwhereas Fermat'sisstillunsolvedandacontinuingchallenge.Neverthelessbothtypesofmathematicalcuriositieshaveastronglyaddictivequalityforthewould­be mathematicianwhichexistsonalllevelsfromtriviatothemostinspiringaspects. Inthepasttherealwayswereafewmathematicianswhoeitherexplicitlyorbyimplicationgavespecificideasandchoiceofdirectiontotheworkofothers—menlike Poincaré,HilbertandWeyl.Thisisnowbecomingincreasinglydifficultifnotimpossible.Thereisprobablynotonemathematiciannowlivingwhocanevenunderstand allofwhatiswrittentoday. AvolumewrittenmorethanthirtyyearsagobyEricTempleBell,TheDevelopmentofMathematics,containsanexcellentabbreviatedaccountofthehistoryof mathematics.(PerhapsIlikeit,because,touseG.­C.Rota'slanguage,myworkismentionedthereeventhoughthebookwaswrittenwhenIwasonlytwenty­eight yearsoldanditisarathersmallvolume.Thereismoresatisfactioninbeingmentionedinashorthistorythaninonewhichhastenthousandpages!)ButwhenWeyl wasaskedbyapublishertowriteahistoryofmathematicsinthetwentiethcenturyheturneditdownbecausehefeltthatnoonepersoncoulddoit. VonNeumann,whocouldhaveaspiredtosucharole,admittedtomesomethirty­fiveyearsagothatheknewlessthanathirdofthecorpusofmathematics.Athis suggestiononceIconcoctedforhimadoctoral­styleexaminationinvariousfieldstryingtoselectquestionswhichhewouldnotbeabletoanswer.Ididfindsome,one eachindifferentialgeometry,innumbertheory,inalgebra,whichhecouldnotanswersatisfactorily.(Thisbythewaymayalsotendtoshowthatdoctoralexamshave littlepermanentmeaning.)

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Asformyself,IcannotclaimthatIknowmuchofthetechnicalmaterialofmathematics.WhatImayhaveisafeelingforthegist,ormaybeonlythegistofthegist,ina numberofitsfields.Itispossibletohavethisknackforguessingorfeelingwhatislikelytobeneworalreadyknown,orelsenotknown,insomebranchof mathematicswhereonedoesnotknowthedetails.IthinkIhavethisabilitytoadegreeandcanoftentellwhetheratheoremisknown,i.e.alreadyproved,orisanew conjecture.Thisisasortoffeelingthatcomesfromthewaythequantifiersarearranged,fromthetoneorthemusicofthestatement,sotospeak. Speakingofthisanalogy:Icanremembertunesandamabletowhistlevariousmelodiesrathercorrectly.ButwhenItrytoinventorcomposesomenew"catchy"tune, IfindratherimpotentlythatwhatIdoisatrivialcombinationofwhatIhaveheard.ThisincompletecontrasttomathematicswhereIbelievewithamere"touch"Ican alwaysproposesomethingnew. Collaborationinmathematicsisaveryinterestingandnewphenomenonwhichdevelopedduringthelastseveraldecades. Itisnaturalinexperimentalphysicsthatinvestigatorsworktogetheronthedifferentphasesofinstrumentation.Bynoweveryexperimentisreallyaclassoftechnical projects,especiallyonthegreatmachineswhichrequirehundredsofengineersandspecialistsfortheirconstructionandoperation.Intheoreticalphysicsthisisperhaps notasevident,butitexists,andstrangelyenoughinmathematicsalso.Wehaveseenthatthecreativeeffortinmathematicsrequiresintenseconcentrationandconstant thinkingindepthforhoursonend,andthatitisoftensharedbytwoindividualswhojustlookateachotherandoccasionallymakeafewremarkswhenthey collaborate.Itisnowdefinitelysothateveninthemostabstrusemathematicalquestionstwoormorepersonsworktogetherontryingtofindaproof.Manypapers havenowtwo,sometimesthreeormoreauthors.The

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exchangeofconjectures,suggestingtentativeapproaches,helpstobuilduppartialresultsalongtheway.Itiseasiertotalkthantowritedowneverythought.Thereis hereananalogytoanalyzingagameofchess. Itmaybethatinthefuturelargegroupsofmathematiciansworkingtogetherwillproduceimportant,beautiful,andsimpleresults.Somehavealreadybeenproduced thiswayinrecentyears.Forexample,thesolutionofoneofHilbert'sproblemsabouttheexistenceofalgorithmstosolvediophantineequationswasreallyobtained (notinparalleltobesurebutinsequence)byseveralscientistsinthiscountry,andattheendbyayoungRussian,YuriMatiasevic,whotookthelaststep.Several mathematiciansworkingindependentlyintheUnitedStatesandinPolandbutawareofeachother'sresults,solvedanoldproblemofBanach'saboutthe homeomorphismofhisspaces.Theywereabletoclimboneachother'sshoulders,sotospeak. ItwasafterthepublicitysurroundingtheconstructionoftheatomicbombinLosAlamosthattheexpression"criticalmass"becamecurrentasametaphoricdescription oftherequiredminimalsizeofagroupofscientistsworkingtogetherinordertoobtainsuccessfulresults.Iflargeenough,thegroupproducesresultsexplosively. Whenthecriticalmassisreached,duetomutualstimulationthemultiplicationofresults,likethatofneutrons,becomesexponentiallylargerandmorerapid.Before suchamassisattained,progressisgradual,slowandlinear. Othervariationsintheworkinghabitsofscientistshavebeenslower.Themodeoflifeintheivorytowerworldofsciencenowincludesmorescientificmeetings,more involvementingovernmentalwork. Asimplebutimportantthinglikeletter­writinghasalsoundergoneanoticeablechange.Itusedtobeanart,notonlyintheworldofliterature.Mathematicianswere voluminousletterwriters.Theywroteinlonghandandcommunicatedatlengthintimateandpersonaldetailsaswellasmathematicalthoughts.Todaytheavailabilityof secretarialhelp

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renderssuchpersonalexchangesmoreawkward,andasitisdifficulttodictatetechnicalmaterialscientistsingeneralandmathematiciansinparticularexchangefewer letters.InmyfileoflettersfromallthescientistsIhaveknown,acollectionextendingovermorethanfortyyears,onecanseethegradual,andafterthewar accelerated,changefromlong,personal,handwrittenletterstomoreofficial,dry,typewrittennotes.Inmycorrespondenceofrecentyears,onlytwopersonshave continuedtowriteinlonghand:GeorgeGamowandPaulErdös. ChenNingYang,theNobelprizephysicist,tellsastorywhichillustratesanaspectoftheintellectualrelationbetweenmathematiciansandphysicistsatpresent: Oneeveningagroupofmencametoatown.Theyneededtohavetheirlaundrydonesotheywalkedaroundthecitystreetstryingtofindalaundry.Theyfounda placewiththesigninthewindow,''LaundryTakeninHere."Oneofthemasked:"Mayweleaveourlaundrywithyou?"Theproprietorsaid:"No.Wedon'tdo laundryhere.""Howcome?"thevisitorasked."Thereissuchasigninyourwindow.""Herewemakesigns,"wasthereply.Thisissomewhatthecasewith mathematicians.Theyarethemakersofsignswhichtheyhopewillfitallcontingencies.Yetphysicistshavecreatedalotofmathematics. Insomeofthemoreconcretepartsofmathematics—forexampleprobabilitytheory—physicistslikeEinsteinandSmoluchowskihaveopenedcertainnewareaseven beforemathematicians.Theideasofinformationtheory,ofentropyofinformationanditsroleingeneralcontinuumoriginatedwithphysicistslikeLeoSzilardandan engineer,ClaudeShannon,andnotwith"pure"mathematicianswhocouldandoughttohavedonesolongbefore.Entropy,apropertyofadistribution,wasanotion originatinginthermodynamicsandwasappliedtophysicalobjects.ButSzilard(inverygeneralterms)andShannondefinedthisnotionforgeneralmathematical systems.True,NorbertWienerhadsomepartintheoriginofitandwonderfulmathematicians

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likeAndreiKolmogorofflaterdeveloped,generalized,andappliedittopurelymathematicalproblems. Inthepastsomemathematicians,Poincaréforexample,knewalotofphysics.Hilbertdidnotseemtohavetoomuchtruephysicalinstinct,buthewrotevery importantpapersaboutthetechniquesandthelogicofphysics.VonNeumannknewagooddealofphysicstoo,butIwouldsaythathedidnothavethephysicist's naturalfeelingforandrecoursetoexperiment.Hewasinterestedinthefoundationsofquantummechanicsaslongastheycouldbemathematized.Theaxiomatic approachtophysicaltheoriesistophysicswhatgrammaristoliterature.Suchmathematicalclarityneednotbeconceptuallycrucialforphysics. Ontheotherhand,muchoftheapparatusoftheoreticalphysicsandoccasionallysomeprecursorideascamefrompuremathematics.Thegeneralnon­Euclidean geometriespropheticallyenvisagedbyRiemannashavingfutureimportanceforphysics,camebeforegeneralrelativity,andthedefinitionandstudyofoperatorsin Hilbertspacecamebeforequantummechanics.Thewordspectrum,forexample,wasusedbymathematicianslongbeforeanybodywouldhavedreamedofusingthe spectrumrepresentationofHilbertspaceoperatorstoexplaintheactualspectrumoflightemittedbyatoms. Ihaveoftenwonderedwhymathematicianshavenotgeneralizedthespecialtheoryofrelativityintodifferenttypesof"specialrelativities,"sotospeak(notintothe presentlyknowngeneraltheoryofrelativity).Iamsurethereareother"relativities"possibleingeneralspaces,yethardlyanythinghasbeenattemptedby mathematicians.Endlesspapersexistonmetricspacesgeneralizingtheordinarygeometrywithoutthedimensionoftimeinit.Putintimeandspacetogether,and mathematiciansstayout!Topologistsstaywithspatialspaces;theyhavenotconsideredideaswhichwouldgeneralizethefour­dimensionaltime­space.Thisisvery curioustome,epistemologicallyandpsychologically.(IcanthinkofonepaperbyvanDan­

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zig,whichspeculatesphilosophicallyaroundthenotionoftimetopology;hesaysitmightbeasolenoidalvariable.Ilikethis,butclearlyoneshoulddomuchmore imaginativeworkwithtime­likespaces.) Asiswellknown,thetheoryofspecialrelativitypostulatesandisbuiltentirelyonthefactthatlightalwayshasthesamevelocityregardlessofthemotionofthesource ortheobserver.Fromthispostulatealoneeverythingfollows,includingthefamousformulaE=mc2.Mathematicallyspeaking,theinvarianceoftheconesoflightlead totheLorentzgroupoftransformations.Nowamathematiciancould,justformathematicalfun,postulatethatthefrequency,forexample,remainsthesame,orthat someotherclassofsimplephysicalrelationisinvariant.Byfollowinglogicallyonecouldseewhattheconsequenceswouldbeinsuchapictureofanot"real"universe. Mathematicsisnowsocompletelydifferentfromwhatitwasinthenineteenthcentury,evenifninety­ninepercentofmathematicianshavenofeelingforphysics.There aresomanyideasinphysicsbeggingformathematicalinspiration—newformulations,newmathematicalideas.Idonotmeantheuseofmathematicsinphysics,but theotherwayround:physicsasastimulantfornewmathematicalconcepts. Contrarytomathematics,inphysicsonecan,inprinciple,keepmoreabreastofwhatisgoingoninresearch.Everyphysicistcanknowthegistofmostofphysics. Thereareveryfewfundamentalproblemsnowsuchastheproblemofthenatureofelementaryparticlesorwhatisthenatureofthephysicalspaceandtime. Inpresent­dayresearchintheoreticalphysics,eventhoughmanyoftheyoungpeopleareveryclever,ingenious,andtechnicallysuperb,theirfundamentalideastendto beorthodox,andonthewholeonlysmallvariationsonwhathasbeendoneareproduced,elaborationofdetails,andcontinuationalonglinesthatalreadyhavebeen started.

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Perhapsthishasalwaysbeensoandreallynewideasareexceptional. Sometimes,halfinjest,toneedlecontemporaryyoungphysicistfriendswhospendalltheirtimeexaminingafewverystrangeparticles,Itellthemthatitisnot necessarilythebestwaytogetnewinspirationaboutthefoundationsofphysicsandtheschemeofthingsinspaceandtime. Ofcourse,itisnotapreciseproblemorrecognizedassuch,butwhattomymindisafirstquestioninphysicsiswhetherthereexistsatrueinfinityofstructuresgoing downintosmallerandsmallerdimensions.Ifsoitwouldbeworthwhileformathematicianstospeculateonwhetherspaceandtimechange,evenintheirtopology,in smallerandsmallerregions.Wehadinphysicsanatomisticorfield­structurebase.Iftheultimaterealityconsistsofafield,thenitspointsaretruemathematicalpoints andindistinguishable.Thereisapossibilitythatinrealitywehaveastrangestructureofinfinitelymanystages,eachstagedifferentinnature.Thisisafascinatingpicture whichbecomesmorephysicalandnotmerelyaphilosophicalconundrum.Recentexperimentsshowdefinitelytheincreasingcomplicationofstructures.Inasingle nucleonwemayhavepartons,asFeynmancallsthem.Thesepartonsmaybethehypotheticalquarksorotherstructures.Therecenttheoreticalattemptsnolonger explaintheexperimentalmodelsbysimplequarks,butonehastoinvolvecoloredquarksofdifferenttypes.Perhapsonehasreachedapointwhereitmightbe preferabletoconsiderthesuccessionofstructuresasgoingonadinfinitum. Theoreticalphysicsispossiblebecausetherearemanyidenticalornearlyidenticalcopiesofobjectsandsituations.Ifonetakestheuniversebydefinitionasonlyone (eventhoughitistruethatgalaxiesresembleeachother)andtheworldasawholebeingone,thequestionsaskedaboutthecosmosasawholehaveadifferent character.Thestabilitywithrespecttoaddingafewmoreelementstoanalready

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largenumberisnolongerguaranteed.Wehavenowaytoobserveorexperimentwithanumberofuniverses.Thereforeproblemsofcosmologyandcosmogonyhave adifferentcharacterfromthoseofeventhemostfundamentalphysics. Sciencewouldnotbepossible,physicswouldnotbeconceivable,iftherewasnotthissimilarityoridentityofvastnumbersofpointsorsubsetsorgroupsofpointsin thisuniverse.Allindividualprotonsseemtoresembleeachother,allelectronsseemtoresembleeachother,theattractionbetweenanytwocelestialbodiesseemsto besimilar,dependingonlyondistanceandmass.Sotheroleofphysicsappears,interalia,todividetheexistinggroupingsintoentitiesofwhichthereareverymany examplesthatareisomorphicoralmostisomorphictoeachother.Thehopeforphysicsliesinthefactthatonecanalmostrepeatsituations,orifnotexactlyrepeat,the additionofoneormoresmallchangesmakesrelativelylittledifference.Whethertherearetwentyortwenty­twobodiesdoesnotmaketheirbehaviorchangeradically. Abeliefinsomefundamentalstability!Somehowthehopeistodescribephysicsintermsofsimplerentitiesandidentityofpartsbysomekindofunionorcounting. Forexample,physicistsbelieved,atleastuntilrecently,thatifonehadmanypoints,thebehavioroftheirmasscouldbeexplainedbytwo­bodyinteractions—this meansaddingupthepotentialsbetweenanytwobodies.Otherwise,ifeverytimethroughaddingafewbodiesonechangedthebehaviorofthewholesystem,there wouldbenoscienceofphysics.Thispointisnotsufficientlybroughtoutinphysicstextbooks. Onecanrelatethenotionofentropytothenotionofcomplexity,ifonedefinesthedistancebetweentwoalgebraicalstructuresandthetotalworknecessarytoprovea statementoratheoremasenergy.Resultsexiststatingthatingivensystemsinordertoprovesuchandsuchformulaoneneedssomanysteps.Theminimumsufficient numberofstepscanbedefinedasananalogueofworkorenergy.Thisisworththinkingabout.Tomakeasensibletheoryofit

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requireserudition,imaginationandcommonsense.Thereisnoaxiomaticsystemevenforthepresentlyestablishedbodyofphysics. Justasinpuremathematics,intheoreticalphysicswecanseeadichotomybetweenthegreatnew"unexpected"ideasandthegreatsynthesesofestablishedtheories. Suchsynthesesareinasensecomplementaryoropposedtothenewconcepts.Theysummarizeprevioustheoriesinanon­obviousway.Letmeillustratethis distinction:thespecialtheoryofrelativityisaprioriaverystrangeandmysteriousconcept.Itinvolvesanalmostirrationalinsightandanaprioriimplausibleaxiom basedontheexperimentalfactthatlightvelocityseemstobethesameforamovingobserverfromafixedemissionpointorviceversa.Whentheemissionpoint movesawayortowardtheobserverthevelocityoflightrelativetotheobserveristhesamenomatterwhattherelativevelocityis.Fromthisaloneagreattheoretical edificewasbuilt,aphysicaltheoryofspaceandtimewithsomanysurprising—andaswenowknow—technologicallyshatteringconsequences. Quantumtheoryinvolvessimilarly,inaway,anapriorinon­intuitiveorunexpectedsetofconcepts. Maxwell'stheoryofelectromagnetismwouldbeanexampleofagreatsynthesis.Itcameafteragreatnumberofexperimentalfactsweredevelopedwhichwere perhapsnotsostrangetotheirfirstdiscoverers.Thetheorythatexplainstheseobservationalfactsinonesetofmathematicalequationsconstitutesoneofthemost impressiveachievementsofhumanthought.Epistemologicallythistheoryisofadifferentnature,orsoitseemstome,fromrelativityandquantumtheorywhichwere, onemightsay,moreunexpected. Inastronomy,therecentobservationaldiscoveriesshowthecontinuingstrangenessofthecosmosinthevarietyofdifferenttypesofstars,ofconglomerationsofstars, ofclusters,galaxies,andstrangenewobjects.Theseincludeneutronstars,blackholes,andotherverypeculiaranduntilrecentlyunsuspectedpropertiesofassemblies ofmatter,and

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enormouscloudsofmolecules,someofthem"pre­organic"intheinterstellarspace.Againthisisanindicationofthestrangenessoftheuniverserelativetoour conceptions,whichwereacquiredbyprevioussetsofobservationsandunderpreviouscanonsoflearningandknowledge. Inphysicsonegetssurprises,too,frequentlynowinthemoretechnologicalorpracticalconsequencesofsomephysicaldiscoveries.Forexample,theapplicationsof theconceptionanddevelopmentofhologramsandtheirusesareveryperplexingatfirst.Similarly,thenewlasertechniquesareingeneralveryimpressive. Therecentdiscoveriesinbiology,revolutionaryandpromisingastheyareintheirintroductionoffantasticnewvistasofthefuturechangeofthemodeoflifeonour earth,haveadifferentepistemologicalcharacter.Iamstruckbythe"reasonableness"ofthearrangementsonwhichlifehasbeenshowntobebased.Thediscoveries ofthewaylivingmatterreplicates,everythingthatfollowedfromCrick'sandWatson'smodels,thenatureofthebiologicalcodeand"toutcequis'yrattache,"asthe Frenchsay,showonthecontraryasortofverycomprehensibleandalmostnineteenth­centurytypeofmechanicalarrangementswhichdonotrequirebasicphysicsfor theunderstandingofhowtheyoperate.Quantumtheoryisimportantfortheexplanationofthephenomenaofthebasicmolecularreaction,thebasisofthe arrangement,butthearrangementsthemselvesseemtobequasi­mechanicaloralmostquasi­engineeringinthewaytheyinvolvetheveryfabricofthelifeprocesses. Onecouldaskwhyisthisso?Whyisitthatourunderstandingofthephysicalworldandperhapstheworldoflivingmatterorourselvesorthepatternofourthoughts, doesnotseemtoproceedoraccruecontinuously?Insteadofalogicaldevelopmentof'steadygrowthweobservediscretequantumstages.Isitthattheworldisreally simpleinitsineffablestructure,butthattheapparatusofthenervoussystemwhichbringsittoconsciousnessorrendersitsunderstandingcommunicablemust,of necessity,becomplicated?

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Isitthatthestructureofourbrainwithallitsneuronsandconnections,anadmittedlyverycomplicatedarrangement,isnotbestsuitedtoadirectdescriptionofthe universe?Orperhapstheotherwayround,realityisonsomeverycomplicatedobjectivescalewhichwedonotyetevenconceiveandwe,inoursimplemindedway, trytogleanitanddescribeitbysimplestepsinsuccessiveapproximationsasDescartesprescribedinhisDiscoursdelaMéthode? (Foramoredetailedconsiderationofthepossibilitiesofafutureroleofmathematicsinbiologicalresearch,IreferthereadertoanarticleIwroteentitled"SomeIdeas andProspectsinBiomathematics."Thetechnicalaspectofthearticleissomewhatbeyondthescopeofthesegeneralremarks,butinterestedreadersmayliketolook atit.) Insocialsciences,alaymanlikemyselffeelsthatthereisnotheoryordeeperknowledgeatthepresenttime.PerhapsthisisduetomyignorancebutIoftenhavethe feelingthatbyjustobservingthesceneorreading,say,TheNewYorkTimes,onecanhaveasmuchforesightorknowledgeineconomicsasthegreatexperts.Idon't thinkthatforthepresenttheyhavetheslightestideawhatcausesthemajoreconomicorsocio­politicalphenomenaexceptforthetrivialitieseveryoneshouldknow. Adevelopmentwhoseeffectwecannotevenestimateandwhichwouldhaveanimpactgreaterbyfar,Ithink,thananyoftheestablishedreligions,willbethe discoveryoftheexistenceofotherintelligencesintheuniverse—perhapsthousandsoflight­yearsawayfromthesolarsystem.Itisentirelypossiblethatthereare waveswhichhavetraveledforalongtimewhichwecouldsuddenlydecipher.Aninklingoraproofofexistencewithoutthepossibilityofcommunicatingtwoways wouldhaveanoverwhelmingeffectonhumanity.Thiscouldhappenverysoon,anditmightcreatepanicor,onthecontrary,anewtypeofreligion. Wehaveallreadaboutflyingsaucersandotherunidentifiedflyingobjects.EdwardU.Condondirectedaverythoroughstudyofthesubject.Mostcaseswereeasily provedto

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beeitherillusions,opticalorotherwise,ornaturalatmosphericphenomena,butthereremainafewcasesofauthenticatedUFO'swhicharemostpuzzling.Takethe groupofMountWilsonastronomerswhowereonawalkandsawaverystrangemeteoricobject,andwhentheyreturnedtotheobservatorysawindicationsofhigh peaksofradioactivity.Therearealsoafewcasesofobjectswhichweresimultaneouslyfollowedvisually,byplaneandbyradar,whichhaveneverbeenexplained. Fermiusedtoask:"Whereiseverybody?Wherearethesignsofotherlife?" Inmyopinion,morethananythingelseitisthenewbiologywhichwillchangethewayoflifeinourworldduringthenexttenorfifteenyears.Discoverieswhichatfirst seemedratherordinaryhavealreadyhadmoreeffectonthecompositionoftheworldthanevenbigwars:newdrugslikepenicillinontheonehand,contraceptiveson theother,havechangedthebalanceofpopulation. Toillustratethefastpaceofdiscoveriesinbiology,Irecentlyheardoftwoimportantdevelopmentsincancerstudiesinthespaceofoneweek.OneisthataMichigan scientisthasdiscoveredavirusinthehumanbreastcancercell.TheotheristhatanexperimentwhichwasactuallyperformedinBoulder,wherethereisaverygood electronmicroscope,resultedinasurprisingnewtechnique.KeithPorterandhisassociateswereabletoproducecellsfromwhichthenucleicouldbeextracted. Theseundamagednucleicanbetransferredtootherdenucleizedcells,soineffectitisanexchangeofnucleibetweencells.Forexample,acancerouscellcanhaveits nucleusremovedandputintoahealthycell.Thenthenewcellmaybecomenormal.Thisismostremarkableandshowsthatsomeinstructionsmaybecomingnotfrom thenucleus,aswasbelieved,butfromthecytoplasm. Inthefuture,newwaystoproduceorreplacefoodwillhavemuchmoreinfluenceontheshapeandmodeofhumanlifehereonearththananypolitico­socio­ economicdevelop­

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mentsinthepresentsenseofthewords.Allthismaybeobvious,butsometimesobviousthingsneedtoberepeatedoverandoverbeforetheyarerealized.Theworld willbesodifferent.IamremindedofabookwhichwaspublishednotsolongagounderthetitleTheNextMillionYears.Whatlackofimaginationitshows! Gamow'sinterests,vonNeumann'sforesight,Banach'sandFermi'swork,amongothers,haveallcontributedtoenlargetheaspectoftoday'sscience,andenormously widentheperspectivesofphysicsandmathematics.Itisamarvelthatsomanynewvistasandachievementsareduetosuchafortuitousandfortunateconfluenceofall thedivisionsofscience.

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POSTSCRIPTTOADVENTURES FrançoiseUlam WhenScribner'surgedStantowritehismemoirsheagreedtogiveitatryandin1972hetookasabbaticalfromtheUniversityofColoradotodevotehimselftothe task.Forayear,whilewetraveledtotheEastandtoParis,hedictatedreminiscenceswhichItape­recordedandtranscribed.BackinBoulder,Stanreturnedtohis universitydutiesandIeditedandassembledthegiantjigsawpuzzlethetranscriptshadbecome,untiltherewasadraftforhimtolookatandaddafewconnecting sentenceshereandthere.Thebookappearedin1976underthetitleAdventuresofaMathematician.(Stanwantedtochangethetitleto''Misadventures,"buton thathewasoverruled.) "Adventures"isStan,pureStan,allStan—eventhoughhehardlywrotealineofit—forIscrupulouslyrefrainedfrominjectingmyselfintoit.Itisabouthisprofessional lifeandthescientifictimeshelivedin.Itstoneispersonalbutnotintimate,factualmorethananalytical.Inthespiritofthebook,Iwanttocomplementhisstorywitha fewimpressions,perceptions,andmemoriesofmyown,toweavealongaloosethreadoftimemygradualdiscoveryofthemanhewasandthelifeintowhichhe transportedme. StanandImetinCambridgeatthehomeofamutualPolishfriend.IwasaFrenchexchangestudentofliteratureatMountHolyokeCollege;hewasalreadyarather well­knownyoungPolishmathematicianandalectureratHarvard.Inthecollegestyleofthetimes,itwas"HolyokegraduatestudentmeetsHarvardprofessor."But theyearwas1939andWorldWarIIwasengulfingEurope.CastawaysfromtheruinsoftheOldWorld,wewerebroughttogetherontheshoresoftheNew.Iknew atoncethathewassomeonequiteoutoftheordinary,andhebecamethefocusofmylife.Inreturnhegavemeafrontrowseattohisall­absorbingworldofscience

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andscientists.Andwhataworldthatturnedouttobe.Weweremarriedin1941inWisconsinandwelivedinNewMexicothebetterpartofourlives. WhathadfirstattractedmetoStanwasthat,likealleducatedPoles,hewasaFrancophileandspokeafluentFrenchrollingwithSlavicr's,andthatevenduringthese anxioustimesinthedarkestdaysofthewar,heseemedtohaveabsoluteself­confidenceandunflinchingoptimism.HewassuretheAllieswouldwin,thoughthe questionofwhenremained.Iwasalsodazzledbythebreadthofhiscultureandhisencyclopedicmemory. Withhisunusuallooksandmagneticgreeneyes,healwaysseemedtostandoutinacrowd.IrememberapartyyearslaterwhenGeorgiaO'Keefepointedan imperiousfingerinhisdirectionandexclaimed,"Whoisthatman?" "Thatman"wasamaverick,astudyincontrastsandcontradictions,aproudPolewhodidnotkowtowtoanyoneandanassimilated,agnosticJewveryconsciousof hisethnicity.ThistranslatedlaterintohisbecomingveryinvolvedwithaPolish­Americanculturalorganization,theAlfredJurzykowskiFoundation,andwiththeIsraeli WeizmannInstituteofScience,theonlyboardsheeverenjoyedactivelyservingon.Andhewasalso,likemanySlavs,improvidentandunorganized.Hereadilyleftall practicalmatterstomeinthemanagementofourdailylife.HelovedtobewithpeopleandentertainthemwithhisquickwitandinexhaustiblefundofJewishjokes, buthewasalsoinsularandinsulated,attimessensitiveorinsensitive,perceptiveorimperceptiveofothers.Someoneoncejokinglytoldhimthathesufferedfrom"le complexeduroi"withhisdisregardoftherulesofordinarylife.Lastbutnotleast,hehadthebuildofanathleteandhealthy,earthyappetites.Healsoclaimedhedid notknowthemeaningofthewordtired,andalthoughheenjoyedtennisandhadplayedsoccerinPoland,helikedtoattributehisgoodhealthtohisdisdainfor exercise.Whatheexercisedallthetimewashismind. Foraboveallhewasamathematician,andheimmediatelythrustmeamonghiskind.EarlyonInoticedthatmathemati­

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ciansliveinaworldinaccessibletocommonmortals,andeventoeachotherwhentheybelongtodifferentdisciplines.Theyareaspecialbreedpossessedbyan intensecerebrallife;simultaneouslylivingontwodistinctlevelsofconsciousness,theyareatoncepresentandabletocarryonnormallyandyetareimmersedinthe abstractionsthatformthecoreoftheirlives.Theyarequitedifferentfromthecommunityofphysicistsweweresoontoencounter,whoseemedmuchclosertothereal world.Stanhadaspeciallookinhiseyeswhenpursuingamathematical­thought.Yetnomatterhowabsorbedhewas,heneverseemedtomindbeinginterrupted. Singularlymodestabouthisscientificaccomplishments,hewasneverthelesskeenlyawareofhisownqualities,whichhedescribedas"ablendofmemoryand imagination,thesimpleingredientsofmathematicalorscientifictalent."(Temperedbyahealthydoseofcommonsense,Imightadd.)Healsoseemedtoprefertolet hisimaginationroamthantodoelaboratetechnicalcalculations."Therearesofewofus,andsomanyofthem,"hesaid.Hisoldprofessorandfriend,Steinhaus,told me"C'estl'hommeaumondequiposelemieuxlesproblèmes."(Heisthemanintheworldwhoknowsbesthowtocreateandformulateproblems.)Hispretensethat heneverstrainedhimselfwithhardworkwasapose.Hisseemingly"effortlessluck"andthe"brilliantthoughts''hewasknownfordidnotappearoutoftheblue.From hisdictationwhichItookinshorthandinthepast,Isawhowtenaciouslyhereturnedtothesamepointsoverandoveragain,eachtimeprobingalittlefurther. Healsoseemedtoprefertoembarkonnewpursuitswhenevertheyoccurredtohimthantoseeanythingthrough,whetheritwasamovieheenjoyedorascientific point,Iaskedhimaboutthisonceandhereplied,"IneverliketoseeordoanythingtotheendforfearthequalitywillwaverandIwillbedisappointed."Heturned thisintoadisdainforputtinghisworkinthemoreformalandfinalprintedform.Forthathereliedoncollaboratorsor"thoughtprocessors"asIcalledthem,happyto letthemhelpattheirvariouslevelsofexpertise.ToDavidHawkinsitmeant"Stanwoulddramatizea

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topic,suggestapathway,andthenIwoulddothework."Everettsaid,"StantellsmewhattodoandIdoit."Atmylevel,andincreasinglyafterretirementwhenheno longerhadregularsecretaries,itmeantservingashis"livewordprocessor.'' MarcKac,whohadknownhimwhentheywerestudentsinPoland,toldmethat"therewasnotasinglemathematicianwhoreminded[him]ofStan,"forStan,besides havingan"enormousreservoiroforginality,"also"dependedmorethananyoneelse[he]knewontheintellectualstimulationthatcomesfrompeople,eventhoughin 99.999%ofthecaseshewasthegiver."Hesaidthat"toStanmathematicswasamuchwidersubjectthantomostofus,"thathe"couldseetherelevanceof mathematicalideastothingswhichperhapsonewouldnotcallmathematics,"andthathewas"thefirstandprobablytheonlyonewhoreallyexperimentedwith machinestodiscoverinterestingfacts,tostimulateconjectures." HeexplainedthatsinceStancamefroma"culturebasedonleisureanddiscourse,"froma"peculiarkindofPolishexistencewhereyouwereincafesallhoursofthe dayornightdrawingdiagramsonsmallpiecesofpaper,"hewasnotverywellsuitedfortheAmericansystemof"timetablesandsomanyhoursofteaching."And indeedStanhardlyeverconformedtothismorestructuredwayoflife.Heneverbecameanine­to­fiveman,notevenatLosAlamos.Butdespiteacertainnostalgia forhisPolishpast,Stanonthewholethrivedinthiscountry.Heloveditsopenness,dynamism,andscientificaudacity. Wheninthefallof1943JohnvonNeumannrecruitedhimtojointheManhattanProject,Stan'slifetookanabruptturn."AllatonceStanwasconnectednotonlywith weaponsbutwiththatreservoirofthesmartestpeopleintheworld.Thatexcitedhisimaginationandhewouldalsoexcitetheirs,"Kacsaid.AtLosAlamos,his friendshipwithJohnnyandtheforceofhispersonalitysoonpropelledusamongthemostinterestingphysicists.Stanrelishedtheworkatthefrontiersofphysicsand theintensityoftheexchangesbetweenthemembersofthisisolatedcommunity.TheyremindedhimofLwów.And

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thesymbiosisprovedveryfortunate,forStanhadneverconsideredhimselfonlyamathematician.FromthefirsthelovedNewMexico'svistasandthequalityofitsair. Helikedtosayitwaslikechampagne.InretrospectIthinkthatwewereallalittlelight­headedfromitandfromthealtitude. Asforme,wartimeLosAlamos,perchedasitwasonitsforestedmesa,wasastrangecombinationofSwissvillage,constructionsite,andarmypostwithPXand commissarywherethesunalwaysseemedtoshinewhatevertheseason.Clusteredaroundhandsomelogbuildingsandbuiltalongthecontouroftheland,itwas essentiallyOppenheimer'screation:asortofMagicMountain.Lifewasrusticandegalitarian.Themenworkedsidebysidedayandnightbehindfencedareas.The wivesmostlykepthouseandhadbabiesborninP.O.Box1663,SantaFe,NewMexico,includingourdaughter,Claire.Thesebabieswerethenraisedwiththehelp ofHispanicwomenwhospokelittleEnglishandIndianmaidsinnativedresssilentlyglidingondeerskinmoccassins.Themilitarywerepresenttohelpthestrange, internationalgroupofciviliansOppenheimerhadgatheredthere. WhenthewarendedanditseemedthatLosAlamoswouldfold,wepromptlywenttotheUniversityofSouthernCaliforniainLosAngeleswhereStansufferedthe traumaticillnesshedescribes.Convalescingatthebeachwithhisheadstillswathedinbandages,hesoonshowedthathisbrushwithdeathhadlefthimunscathed. BesidesworkingonanobituaryofBanachandtalkingalittlemathematicswithErdös,whowasvisiting,hepassedthetimestandingbyanarrowtable,liningup solitaireaftersolitaire(Canfields).ThisexerciseledtothedevisingoftheMonteCarlomethodforcomputingneutronmultiplication. Ashisstrengthreturned,ratherthanremaininthebackwatersofacademe,StanoptedforapromptreturntoLosAlamos,whichwasnotclosingitsdoorsafterall. FromthelatefortiestotheearlysixtiesthenucleusofthetownexpandedtoothermesasandLosAlamosslowlyevolvedintoanalmostnormalcommunitycomplete withtracthouses,stores,andchurches.Butthecountrysideandtheclimatere­

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mained,anditsscientistswerestillattheforefrontoftheworld'seffortstoadapttothenewatomicage.Itwasstillagatheringplaceforthebestthecountryhadto offer,whowerestillblazingtrailsatthefrontiersofscience.ThecoldwarandthescientificandpoliticalcrisesthatsurroundedthedevelopmentoftheH­bomb becamepartofourdailylives,andwecontinuedtoliveinawhirlofactivityatthelabandathome.Stanhadnomoralqualmsaboutworkingonweapons.Hewas interestedinthescientificaspectsoftheresearchanddidnotseeanythingwronginthat. InthesamesenseasthewartimeA­BombhadbeenOppenheimer'sdoing,thepostwareH­bombwasessentiallyTeller's,exceptthatStanhadtoshowhimhowto goaboutit,andLosAlamoshadtobuilditwithouthim. AndhereIhaveafootnotetoaddtoStan'sdescriptionoftheeventsthatledtoitsbuilding.Stanwasacquaintedwiththefusionproposals,forduringthewarhehad beennominallyinTeller'sgroup.GraduallyhedevelopedahunchthattheSuperTellerenvisagedwasnotverypractical,andwhenin1949Trumanorderedthe developmentofthatsuperpowerfulweapon,hesetouttoverifyhishunchwhiletheelectronicmachineswerebeingbuilt.First,heworkedjustwithEverett,then,with anaddedbevyofyoungwomenwhohadbeenhastilyrecruitedtogrindmanuallyonelectriccalculatorstheoperationsnowroutinelyperformedbycomputers.I becameoneofthem,eventhoughIunderstoodnothingofthemathematicsinvolved.Weboretheglamorousnameof"dataanalysts."ThisjoballowedmeinsideT­ Division,atthetimetheinnersanctumofthelab,where,asafriendlaterput,Icould"getthesmellofthehive." IwaswellplacedtowatchhowpersonallyTellertookthefactthatStanandEverettwerethefirsttoblowthewhistlewiththeircrudecalculations.EverydayStan wouldcomeintotheoffice,lookatourcomputations,andcomebackwithnew"guestimates,"whileTellerobjectedloudlyandcajoledeveryonearoundinto disbelievingtheresults.Whatshouldhavebeenthecommonexaminationofdifficultproblemsbecameanunpleasantconfrontation.Moreagreeablewasthemid­

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morningcoffeehourwherethegreatandnotsogreat—includingthis"dataanalyst"—gatheredtodiscussscientificquestionsthatcametotheirminds,the"stateofthe universe,"orwheretohikeonSunday,oneofFermi'sfavoritetopics. ThetechnicalandpoliticaldebateswereragingwhenStan,mullingovertheproblems,suddenlycameuponatotallynewandintriguingapproach.Engravedonmy memoryisthedaywhenIfoundhimatnoonstaringintenselyoutofawindowinourlivingroomwithaverystrangeexpressiononhisface.Peeringunseeingintothe garden,hesaid:"Ifoundawaytomakeitwork.""Whatwork?"Iasked."TheSuper"hereplied.''Itisatotallydifferentscheme,anditwillchangethecourseof history." I,whohadrejoicedthatthe"Super"hadnotseemedfeasible,wasappalledbythisnews,andanxiouslyaskedwhatheintendedtodo.Herepliedthathe"wouldhave totellEdward."FearingtheTellermightpounceonhimagain,IventuredthatmaybeheoughttotesthisideaonMarkorBradburyfirst.Hedid,butwenttoTellerthe nextdayjustthesame. Tellersawrapidlywherethenewavenuecouldleadandtheyhastenedtowritetheirwell­knownjointreport.Itisintwoparts,Stantoldme,becauseTellerwas adding—inStan'swords—"aparallelscheme"ofhisownwhichalteredStan'soriginalsuggestion.MyimpressionisthatfromthenonTellerpushedStanasideand refusedtodealwithhimanylonger.HenevermetortalkedwithStanmeaningfullyeveragain. Stanwas,Ifelt,morewoundedthanheknewbythisunfriendlyrejection,althoughIneverheardhimexpressillfeelingstowardsTeller.(Heratherpitiedhiminstead.) Secureinhisownmindthathisinputhadbeenuseful,hewithdrew.Atthatmomentinthefallof1951whenthelab'scollectivemindwasoncarryingoutthe monumentaltaskofbuildingandtestingtheweapon(aswellasonthestruggleagainstTeller'sattacks),weescapedtoHarvardforafewmonthsofwell­deserved change.FromthenonStansteadfastlycontinuedtoavoidbeingdrawnintothepoliticalturmoil,exceptforabrieftestimonyinCongressinfavoroftheTestBan Treaty.

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AftertheHarvardvisitwereturnedoncemoretoLosAlamoswherethemanagement,initswisdom,providedStanwithmorefreedomthandidtheacademicworld. Stanthengraduallydisengagedhimselffromweaponstoplaywiththethennew,miraculoustoolofelectorniccomputingthathadbeenbuiltatthelab,theMANIAC. Johnny'sbrainchild,itwasthefirstcompletelyelectronicmachinewithabuilt­ingeneraloperationsprogram.ThisModel­TofcontemporarycomputersenabledStan toapplyhisskillsinpuremathematicstootherfieldsofscienceandexperiment,yearsaheadofthetimes,withpatternsofgrowth,complexity,nonlinearity,chaos.His researchwasforthemostpartwrittenupinobscureLosAlamosreports.Fewpeoplewereawareofwhathewasdoingoutsideoftheirownspecialty.Remarkably hedidnotseemtocare.AndpartlybecauseoftheMANIAC'seasyaccessibility,fortherestofhislifeStanremainedconnectedwiththelab.NowonderSeinhaus nicknamedus"LosUlamos"whenhecametovisitusinthesixties. Ourlife,however,wasnotallworkandnoplay.Wetraveledonextendedleavestoacademiccommunitieseastandwest,andtoFranceforyearlyvacations.We livedsimplybutcomfortablyon"bathtubrow"(so­calledbecauseduringthewarithadcontainedtheonlybathtubsonthepost),raisingClaire,ourchild.Stanwould showtheneighborhoodchildrenthecratersonthemoonwithhistelescope,orpointatSputniktracingitsorbitthroughthesky,orplayagameofchesswithavisiting chessmaster.WebelongedtoagroupthattookvisitorstothebanksoftheRioGrandeforcandlelitdinnersatEdithWarner'sfamedadobehouse,whichhadno electrictyorrunningwater.Themealswereservedbyherlong­braidedIndiancompanion,Tilano.ThecontrastoftheirtimelessIndianlifewiththetwentieth­century realityontheHillofferedagreatmomentofrelaxation. Ourhousewasacontinualopenhouse.ThepartieswegaveforthevonNeumanns,Gamows,Fermis,Fisk,Rabi,andmanyotherswerelively.Ihadaknackfor mixingandmatchingpeople,keptagoodtable,andStanwasalwaysagreatcatalyst.Withfrequenttripstocongressesandmeetingsin

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Russia,Britain,Israel,orSwitzerland,ourlifewasthatoftoday'sscientificjet­set. WhenIvoicedreservationsaboutstilllivingattheheartofthermonuclearwork,Stanwouldreassuremethatbarringaccidents,theH­bombrenderedwarimpossible. Healsoagreed,however,thatthereweretoomanybombsalready,andhe"didnotbelievethatRussiawouldinvadeWesternEurope,oneofthesupposedreasons forsuper­rearmament."(Inthelightoftoday'shappenings,hewasrightasusual.) Thiswasatimewhenthequestionofscienceandmoralitywastakingonnewimportance.Whenaskedabouttheethicsofscience,Stanwasfirmonthenecessityof pursuingscienceforinquiry'ssake,regardlessoftheconsequences.WhatwouldArchimedesandNewtonhavedone,heargued,iftheyhadworriedaboutthe consequencesoftheirthoughts?Withoutcalculusnomodernsciencecouldhavedeveloped.AdmittedlythingshadchangedsincethedayswhenPoincarécouldsay thatmoralityandsciencewerenotinconflict,forbothaimedatthebettermentofmankind.Thereleaseofnuclearenergyandthepossibilityofgenemanipulationhad enormouslycomplicatedtheproblem.Buthewasquicktopointouttheusefulaspectsofnuclearenergyandthemarvelsofbiologicalengineering,ifusedwisely,and toputtheresponsibilitybackinthepublic'scourtfordecidinghowtousethem.Healsoinsistedthatitwasthescientists'dutytoinformthepublicoftheirdiscoveries. WiththeColdWarwaningandtheBradburyeraending,thepurposeofLosAlamoswasbecomingdimmer,andthelabwassettlingintoroutine.Itwasalsoentering periodstop­heavywithbureaucraticadministrators.Stan,detachedfromitsinnerworkings,foundhimselfchartingaloneliercourse,andwasoutgrowing"government science,"ashecalledit.Hethereforeembarkedonarenewedinvolvementwiththeteachingprofession. Hehadalwaysbeennotedforhislecturingstyle.IntheHarvarddaysthestudentshadnamedhimoneofthemoststimulatingmathprofessors.Iwatchedhimonceon thelecturecircuitgiveoneofhistypicalUlamiantalkswhere,hop­

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pingfromtopictotopic,heexposedagroupofmathmajorstoaglimpseoftheworldofsciencetheycouldnotfathomfromtheirmoreprosaicencounters:afewrods aboutsettheory;anecdotesofhisyouthwhenhedevelopedaninterestinmath;sketchesofLosAlamos,ofvonNeumann;remarksaboutthebombattherootof muchpresent­daytechnology;passingthoughtsaboutGödelandtheimportanceofhisundecidabilityproofs.Hewalkedabout,madeagreatdisplayofglancingathis watch,alwaysfindingthemotjusteinaninformalstreamofconsciousness,neversayinganythingtrite,andsowingseedseverysolightlyinhislisteners'minds. WithClairegrownandoutofthenest,StanofficiallyretiredfromLosAlamosin1967andlaunchedaretirementcareerthatkepthimbusierthaneverwithnew universityduties,invitations,meetings,lectures,andextendedstaysinSantaFe,wherewehadboughtahouse. AttheUniversityofColoradohisgoalhadbeentoawakentheacademicmathematicalcommunitytothegreatpotentialofthecomputerageandto"whatphysicsand biologycoulddoformathematics,"hisparaphraseofthefamousKennedystatement.InBoulderheworethreehats:Headofthemathematicsdepartment(withan assistantchairmantorunthedailydepartmentalaffairs);professorinthecomputingcenter;andprofessorinthebiologydepartmentofthemedicalschoolwherehe lecturedoccasionallyonthespeculationsandmathematicalexplorationsofneurophysiologyandtheworkingsofthebrain,whichhadreplacedtheattentionhehad devotedearliertophysicsandastrophysics. InhisacademicsparetimeatourSantaFepied­à­tierrehecontinuedtousetheadmirableLosAlamoscomputingfacilitiesasadollar­a­yearconsultant.Buthewas sorelymissingthoseofhispeerswhohaddiedbeforetheirtime,andbeginningtofeelremovedfromtheyoungergenerationofscientistswhoarrivedduringour absences.Tocounterthat,whenheretiredfromBoulderheestablishedalooseconnectionwiththeUniversityofFloridaasVisitingGraduateResearchProfessorof Mathematicswhichremainedineffecttotheend.InGainesvilleheconcentratedongivingcolloquia

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andseminarsonnewmathematicalproblemsheverymuchwantedtoassembleintoanewvolumeofproblems.Thebookwasbeginningtotakeshapewhenhedied. Mauldin,hiscollaboratorontheproject,hassincepublishedanabbreviatedversioninamathematicaljournal. Invitationstolectureweregrowingexponentially.IfhehadacceptedthemallhewouldhavebecomelikeErdös,anitinerantperformer.Claiminghewas "nonbiodegradable,"inthefirstsixmonthsof1978,inadditiontoextendedstaysinSantaFeandGainesville,StangavetalksinParis,Montreal,Vancouver,New York,Washington,Denver,SantaBarbara,Parisagain,andWarsaw,pridinghimselfthatheneverexactlyrepeatedhistalks.Ontheroadheexperiencedachargeof adrenalin.Acertainletdowncameafterthereturnhome:catchingupwithmailandotherobligations,fatigue,increasinglyfrequentdigestiveupsets.Towardsthevery endthestrainwasbeginningtotell. Then,onMay13,1984,attheageof75,while,inErdös'words,"stillabletoproveandconjecture,"onadayhehadcontentedlyreturnedhomefromLondon,Stan hadanunexpectedseizure,collapsed,anddied.Thissparedhimthedisintegrationhefeared,andmuchasIstillmisshim,italsosparedmethedistressofwatching himgrowold.Inhisusualwayofmakinglightofseriousmatterstorenderthemmorepalatable,heusedtosay"thebestwaytodieisofasuddenheartattackorto beshotbyajealoushusband."Hehadthegoodfortuneofsuccombingtothefirst,thoughIbelievehemighthavepreferredthesecond. InabriefmemorialeulogyDavidHawkinssaid:"Peoplewhodonotmuchlivecannotmuchdie.Stanlivedverymuch.Thosewholiverichlyhavemanylinkagestothe world...Theirlivesarewovenintotheworld'sfabric,itslatticeofassociations.Whentheyleavethereisabigholeinthelattice,atearinthefabric.Theseholesor tearsremain...andthatsimplefactisthehumansourceofallourconcernsforimmortality." FrançoiseUlam SantaFe,1990

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SELECTEDBIBLIOGRAPHY INDEX

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SELECTEDBIBLIOGRAPHY Books BYS.M.Ulam

CollectionofMathematicalProblems.NewYork:Interscience,1960. Sets,NumbersandUniverses.Cambridge,Mass.:TheM.I.T.Press,1974. EDITEDANDTRANSLATEDBYS.M.ULAM

TheScottishBook:ACollectionofProblems.TranslatedfromanotebookkeptattheScottishCoffeeHousefortheuseoftheLwówSection,PolishMathematical Society.LosAlamos,N.Mex.:LosAlamosScientificLaboratory,1957. BYMARKKACANDS.M.ULAM

MathematicsandLogic.NewYork:Praeger,1968. Articles BYS.M.ULAM

"CombinatorialAnalysisinInfiniteSetsandSomePhysicalTheories."ReviewofSocietyofIndustrialAppliedMathematics,vol.6(1964),pp.343–355. "GamowandMathematics."InGamowMemorialVolume.Boulder,Colo.:UniversityofColoradoPress,1972. "IdeasofSpaceandSpaceTime."Rehovoth,Israel:WeizmannInstitute,winter1972–73. "Infinities."InTheCopernicanVolumeoftheNationalAcademyofSciences,Cambridge,Mass.:TheM.I.T.Press,1974. "JohnvonNeumann,1903–1957."BulletinofAmericanMathematicalSociety,vol.64,no.3,pt.2(1958),pp.1–49. "OntheMonteCarloMethod."InProceedings,SymposiumonLarge­ScaleDigitalCalculatingMachines,September1949.Cambridge,Mass.:HarvardUniversity Press,1951.

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"SomeIdeasandProspectsinBiomathematics."AnnualReviewofBiophysicsandBioengineering,vol.1(1972),pp.227–292. BYP.R.STEINANDS.M.ULAM

"ExperimentsinChessonElectronicComputingMachines."ComputersandAutomation,vol.6,no.9(1957),pp.14–18.

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INDEX A Abel,NielsHenrik,207 AEC,seeAtomicEnergyCommission Agnew,Harold,192 AirForceSpaceCommittee,255,256 Alembert,Jeand',272 Alexander,James,70,244 Alexandroff,Paul,46,66,68 Alvarez,LuisW.,211 Anderson,SenatorClintonP.,72,254 Archibald,RaymondC.,100 Archimedes,104,274,290 Argo,HaroldandMary,153 Aron,Françoise,seeUlam,Françoise Aronszajn,Nachman,36 astronomy,cosmology,16ff,203ff,221,255,258,299 Atkinson,GeoffreyS.,149 atomicbomb,5,142,146,149f,152,159,170,188f,209,211,221f atomicenergy,189,237; seealsonuclearenergy AtomicEnergyCommission,193,210ff,221,224,228,231,237,240,243,253; GeneralAdvisoryCommitteeof,216,220 Auerbach,Anna,seeUlam,Anna Auerbach,Herman,42 Auerbach,Karol,56 automata,32,82,96,98,241f,285; seealsoelectroniccomputers Ayres,WilliamL.,107 B Bacher,RobertF.,150 Baker,Nicholas,seeBohr,Niels ballistics,137,231 Banach,Stefan(1892­1945),6,22,26,31ff,38,40,49,54f,65,107,180,203,245,293,303 Bardeen,John,86 Bell,EricTemple,291 Bell,George,259 Bellman,Richard,131f Bergman,Stefan,119 Bernouillilawoflargenumbers,73 Bernstein,Dorothy,131 Bernstein,Felix,27 Besicovitch,Abraham,59f Bethe,HansAlbrecht,143,147,149f,152,156,162,172,191ff,216,260,263 Beyer,William,268 Bigelow,Arthur,229 biology,203ff,242,259f,300ff Birkhoff,Garrett,66f,84,89f,92 Birkhoff,GeorgeDavid(1884­1944),6,74ff,82,85,89,90,

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Birkhoff,GeorgeDavid(cont.)95,98f,101,113,117,120ff Blair,Clay,211,236 Bloch,Felix,161 Bochner,Salomon,66,69,73 Bohr,Niels(1885­1962),81,161f,168f,286 Boltzmann,Ludwig,56,85 Bolzano,Bernard,282 Boole,George,144 Borel,Emile,282 Borsuk,Karol,40f,57,68,270 Boulder,Col.,104,205,250,266,302 Bradbury,Norris,192,211f,215,220f,223,249 brain,workingsof,12ff,19,52,78,97,175f,180ff,199,236,241f,244,274,276,279,284,289,301 Breit,Gregory,124,132,150 Breton,HalandHattievon,174f,179 Brillouin,Léon,132 Brillouin,Stéfa,132f BritishMission,228,250 Broglie,LouisVictorde,165 Brown,Harold,255 Budenny,MarshallS.,15 Burks,Arthur,241,285 Bush,Vannevar,251 C CaféSzkocka,seeScottishCafe Calkin,JohnW.,169,177,190,209 Cambridge,England,38,58ff Cambridge,Mass.,29,88,94,114,116,119,122,138,205,242 Cantor,Georg(1845­1918),22,29,105,263,272,282 Capablanca,José,19 Caro,108f Cartan,Elie(1869­1951),57 Cauchy,Augustin(1789­1895),85 Cayley,Arthur(1821­1895),264 CERN(CentreEuropéendeRechercheNucléaire),261f Chadwick,SirJames,161 Chandrasekhar,Subrahmanyan,60f Cherwell,Lord(FrederickA.Lindemann),229,251 chess,18,23,35,42,55,120,179,202,280 Churchill,SirWinston,229,251 Chwistek,Leon,29f Coble,ArthurB.,74 Cohen,Leon,130 Cohen,Paul,55,96,283,285 computingmachines,seeelectroniccomputers Conant,JamesB.,251 Condon,Edward(1902­1973),104,203,267,301 CongressionalSpaceCommittee,254 Conway,JohnB.,285 cosmology,seeastronomy Crick,SirFrancisH.,204,241,260,300 Cushing,Harvey,175 Czech,Edward,107 D Dan,Klara,seeNeumann,Klarivon Danzig,J.van,295 Darboux,Gaston(1842­1917),33,58 Davis,RobertR.,169 DelSasso,LewisA.,244 Descartes,René(1596­1650),194,206,301 Dickstein,Samuel(1851­1939),45 Doolittle,GeneralJamesH.,257

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Doyle,SirArthurConan,59,145 Dykstra,ClarenceA.,126 Dyson,Freeman,256 E Eddington,SirArthurS.(1882­1944),60,62 Eilenberg,Samuel,66,68,134 Einerson,Ivar,86 Einstein,Albert(1879­1955),4,6,18,47,62,73,81,108,112,157,167,184,205,223,272,294 Eisenhower,PresidentDwightD.,243,251,253,255 electroniccomputers,4,13,96,137,155,181f,197f,200f,212f,215f,218,225f,229,241,244,257,264,284ff; seealsoautomata Eliot,CharlesW.,87 Elliott,Josephine,215 Erdös,Paul,53,54,60,131,134ff,178f,294 Ernst,Marcin,16f Estreicher,Karol,30 Euler,Leonhard(1707­1783),20,58,104 Evans,CerdaandFoster,213 Evans,Herbert,130 Everett,C.J.,31,104,124,129ff,160,195,213ff,219f,225,253 evolution,205 F Feinsinger,NathanP.,125 Fejer,Leopold(1880­1959),110 Fermat,Pierre(1601­1665),48,278,291 Fermi,Enrico(1901­1954),6,152,157,161ff,172,184,191,193,215,217ff,223,225ff,233ff,243,245,257,303 Fermi,Laura,234,236f,282 Feynman,Richard,77,150,152f,156ff,162,169,221,252,297 Fisk,JamesB.,86,193 fissionbomb,seeatomicbomb Flanders,DonaldA.,169 Ford,Kenneth,220 Ford,ColonelVince,255 Foucaultpendulum,230 Fowler,WilliarnA.,257 Frankel,Stanley,153,184f,213 France,Anatole,23,123,224 Freud,Sigmund,18,22 Frisch,DavidandRose,144 Frisch,Otto,6,162 Froman,DarolK.,219 G Galois,Evariste(1811­1832),207,277 Gamow,Barbara,205 Gamow,George(1904­1968),100,151,202ff,211f,231f,266f,270,294,303 Gamow,Rho,233 GardnerSpaceCommittee,256 Gardner,Trevor,255ff Gaulle,GeneralCharlesde,251 Gauss,KarlFriedrich(1777­1855),231,289 Getting,Ivan,86 Goad,Walter,259 Gödel,Kurt,55,76,80,283 Goldstein,Max,218 Gorman,ElizabethS.,244 Gould,Gordon,259 Graustein,WilliamCaspar(1888­1941),75,89 Greenstein,JesseL.,257 Griggs,David,257 Grossman,M.,57 Guerlach,Henry,86 Gurney,HerbertW.,267 Guttierrez,J.J.,168

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H Hadamard,Jacques(1865­1963),121f Halpern,Ada,45 Hanfmann,George,87 Hanley,Professor,125,130 Hardy,GodfreyH.(1877­1947),58,60,94 Harlow,Francis,196 Harvard,5,61,71,75,82,84,86,88f,91f,95,100,112f,116ff,122f,125,128,137,147,191,223,231,260 Hausdorff,Felix,55 Hawkins,David,153,158ff,174,190,214,266,281 Hawkins,Frances,161 Heidegger,Martin,264 Heisenberg,Werner,165,221 Henderson,LawrenceJ.(1878­1942),86,88,118 Herblock(HerbertBlock),251 Hermite,Charles,58 Hilbert,David(1862­1943),6,45,57,76,81f,282,289,291,293,295 Hill,Albert,257 Hinton,Joan,144 Hirniak,48f Hitler,Adolf,35f,263 Hoffmann,Frédéricde,211,220 Homans,George,88 Hopf,Heinz,57 Houtermans,Fritz,149 Hurewicz,Witold(1904­1956),116 Hurwitz,Henry,Jr.,145 Husserl,Edmund,264 hydrogenbomb,5,109,131,152,184,195,209ff,219ff,234,236f,251 Hyers,Donald,124,131,173,186 I ICBM(IntercontinentalBallisticMissile),231; vonNeumannICBMCommittee,241,255,257 implosion,153f Infeld,Leopold,61f Ingraham,Mark,120,127,134,144 InstitutPoincaré,57f InstituteforAdvancedStudies(Princeton),4,47,66,68ff,73f,134,238 J Janiszewski,Zygmunt(1888­1920),25,38 JimenezyFlores,56 Johnson,PresidentLyndonB.,251,254f K Kac,Mark,43f,96,255,268ff Kaczmarz,Stefan,26f Kárman,Theodorevon(1881­1963),70,79,109 Kennedy,PresidentJohnF.,86,89,251,253,256f Kister,James,201 Kistiakowski,GeorgeB.,143,169,253 Kleene,Steven,124 Knaster,Bronislaw,40,44,46 Kolmogoroff,AndrejN.,160,295 Konopinski,Emil,149f,152,169 Kowalevski,Gerhardt,19 Kruskal,Martin,227 Kuratowski,Kazimir,25ff,35f,38,40f,44,46f,55,65f,75,107 L Lagrange,ComteJosephde(1736­1813),104 Langer,Lawrence,169 Langer,William,134,173 language,influenceofonthought,275

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Laplace,PierreSimonde(1749­1827),58,159 Lauritsen,CharlesC.(1892­1968),257 Lawrence,ErnestO.(1901­1958),161,170,211 Lax,Peter,228 Lebesgue,Henri(1875­1941),282 Lefschetz,Solomon(1884­1972),68f,132 Lehmer,D.H.,117 Leibnitz,GottfriedWilhelm(1646­1716),104 Leonard,WilliamEllery,126 Leray,Jean,68 Lespinasse,Juliede,272 Levin,Harry,86 Levinson,Norman,94 Lindemann,F.A.,seeCherwell,Lord Livermore(Calif.)Laboratory,221,255 Loeb,Carl,Jr.,115 Lomnicki,Antoni,27 Lomnicki,Zbigniew,27,39 Long,Earl,169 Longmire,Conrad,221 Lorentz,HendrykA.,296 LosAlamos,3ff,34,49,61,94,109,131,138,141,144ff,148ff,155ff,160ff,164ff,172ff,177,184ff,188,190ff,195f,199,201,206,209,211ff,220 ff,225,227ff,237,239,249ff,254f,259f,264,266f,269,293 Lotka,AlfredJames,160 Lowell,AbbottLawrence,88 Lowes,JohnLivingston,86 Lwów,4,9f,14,18,23,27,29,32,35ff,39ff,43f,47,50f,65,67f,73,93,99,106ff,116,131,133,269,282 LwówPolytechnicInstitute,24,26,31,47 M McKibben,Joseph,144 Madison,Wisconsin,120,122f,125f,128ff,136ff,144f,173,195 ManhattanProject,145,172; seealsoLosAlamos MANIAC,seeelectroniccomputers Manley,John,249 Marcinkiewicz,Jozef(1910­1940),92,93 Mark,J.Carson,171,193,219,250,266 mathematics,25ff,52ff,65ff,84,90ff,103ff,121ff,128,135,268,273ff algebra,121,130,282,292 algebraicgeometry,134,283 algorithms,280,293 analysis,26,40,98,268 Banachspaces,38,100 branchingprocesses,142,153,159,214 categorytheory,46,85 combinatorialanalysis,134,264,268,282 complexity,298 complexnumbers,278 continuumhypothesis,21,29,196 differentialequations,276 differentialgeometry,291 diophantineequations,293 ergodictheory,85,95,98 Fermat'stheorem,278,291 Fourierseries,transforms,92,96,269 foundationsofmathematics,75,268,284 functionspaces,31,110,137 games,32,242,268,281f,284f,287

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mathematics(continued) grouptheory,30,129,207 infinities,282f,297 informationtheory,160,294 logic,30,75,130,183,259,268 mathematicalphysics,148,192,287 measuretheory,47,55,65,131,282 multiplicativeprocesses,159,213(seealsobranchingprocesses,above) numbertheory,58,92,96,134,268,270,291 numericalanalysis,264 probabilitytheory,27,99,195f,270,282,284f,294 Riemannhypothesis,134,279 settheory,25,27ff,40f,43,65,73,75,96,105,121,268,282 topology,40f,43,66f,103,105,268,270,282f transfinitenumbers,21 undecidability,283 Matiasevic,Yuri,285,293 Matthias,Bernd,254 May,Karl,23,229 Maxwell,JamesClerk,299 Mayer,Maria,110 Mazur,Stanislaw,26,31ff,38,42,50,53,99,104,107,129,270 Mazurkiewicz,Stefan(1888­1945),25,40 memory,12ff,18,54,179ff,199,236 Menger,Karl,56 Menkès,Zygmund,115 meteorology,weathercontrol,185 Metropolis,Nicholas,153,169,177,184f,209,213,225,228,237 Metzger,21 Michael,AristotleD.,177 ''Mike"test,220f Monge,Gaspard,58 MonteCarlomethod,137,196,198f,232 Montgomery,Dean,134 Moore,CharlesN.,283 Morgan,J.P.,231 Morgenstern,Oskar,160,242 Morison,SamuelEliot,86 Morse,Marston,73 Mycielski,Jan,268 N Nabokov,Vladimir,14,124,184 NASA(NationalAeronauticsandSpaceAdministration),254 NavalResearchLaboratory,232 Neumann,Johnvon(1903­1957),4,5,35,44,49,60,65,73,172,188f,207,231,252,291,295,303 AECCommissioner,231,237f,239 associationwithSMU,65ff,100ff,107ff,136f,191,193f,229ff andcomputers,13,184f,198ff,229f,286 inEurope,23,66ff,107ff illnessanddeath,238ff LosAlamosconnection,141ff,146f,153f,161,162,166f,222ff atPrinceton,4,65f,69ff,75ff,95ff Seealsoautomata,electroniccomputers,ICBM Neumann,Klarivon,110,191,193,213,229,239f,243 Neumann,Mariettavon,69,102,109,113 Neumann,Marinavon,66,69 Newton,Isaac(1642­1726),16,59,104,167,205

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Nikliborc,Wladislaw,26,35,37,48 Nikodym,Oton,32 Noether,Emmy,46 Novikoff,PeterS.,285 nuclearenergy,224,252,254 nuclearpropulsion,252,255; seealsoOrion;ProjectRover O Oppenheimer,J.Robert(1904­1968),147,151ff,158,162,170,192,211,217,223f,234,237,251 Orionspacevehicle,253,256 Orlicz,Wladislaw,26 Ouspenski,JamesB.,159 Oxtoby,John,84ff P Paley,Raymond,93 Pasta,John,226ff,257 Pauli,Wolfgang,45,221,261 Perlman,Selig,125 Perrin,Francis,251 Peterson,Donald,259 physics,30,81,98,104,112,132,147ff,162,164ff,189,203ff,216,221,226,233,235,250,261,286,296ff aerodynamics,136f astrophysics,258 complementarity,168 constants,148,169,204,205 electromagnetism,30,279,299 elementaryparticles,112,205,235,261,296 entropy,294,298 hydrodynamics,136f,148,185,196,279 isotopes,233 mechanics,99,137,144,147f,257 mesons,235 neutrons,148,153,159,177,197,213 partons,297 quantumtheory,30,67,77,81,92,97f,112,132,148,165ff,205,245,251,262,279,287,295,299f relativity,18,37,81,105,112,164,221,295f statisticalmechanics,31,132 vacuum,polarizationof,157 Pilsudski,Jozef,15 Placzek,George,158,192 Plateau,Joseph,113 Plato,175,237 Plesch,Arpad,108 Plesch,Janos,108,239 Poincaré,Henri(1854­1912),21,25,45,62,82,98,180,228,275,289,291,295 Polya,George,60,180 Porter,Keith,302 Possel,Renéde,58 Princeton,47,49,62,65ff,72ff,83,102,104,107,110,113,122,132,135,141,202,213,225,229,244 ProjectRover,250,252,254; seealsonuclearpropulsion Proust,Marcel,182 Puck,Theodore,259f Q Quine,Willard,87 R Rabi,IsaacIsidore,111,161,215,223,239 radiation,148,177,216,259,261,267 Rainey,Rupert,175 Rarita,William,152 Richardson,James,100 Richardson,JamesH.,225 Richtmyer,Robert,186,192f,263,266 Riemann,Bernhard,295 Ries,Minna,47 Riesz,Fredrick,110

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rocketpropulsionsystems,254ff; seealsoOrion;ProjectRover Roeberg,Jane,153 Rosen,Henryk,108 Rosenbluth,Marshall,220 Rota,Gian­Carlo,72,104,129,207,263ff,268,291 Rubinowicz,Wojciech,31 Russell,Bertrand,118,137 Ruziewicz,Stanislaw,22,40 S Samuelson,Paul,86 Schauder,Juliusz(1896­1943),39,67 Scherr,Ruben,168 Schmidt,Wolfgang,268 Schouten,JanA.,119 Schrandt,Robert,201 Schreier,Jozef,43,58,78,82 Schriever,GeneralBernardA.,255 Schrödinger,Erwin,165 ScottishBook,The,50f,107 ScottishCafe,33,35,49,93,107 Segré,Emilio,162,164 Selberg,Atle,244 Serber,Robert,152 Shannon,Claude,160,294 Shapley,Harlow,124 Shepley,JamesR.,211,236 Sierpinski,Waclaw(1882­1969),21,25,40,44 Smoluchowski,Marian,294 SocietyofFellows(Harvard),82,84ff,88f,100,113,118 Socrates,175,237 Sommerfeld,Arnold,30 spacevehiclesandspaceexploration,252ff space­time,183f,279,295,297 Stebbins,Joel,144f,126,142 Stein,Myron,201 Stein,Paul,201f Steinhaus,Hugo(1887­1972),21f29,32,35,41ff,50f,58,96,107,262,268f Sternbach,Ludwig,43 Stone,Harlan,92 Stone,Marshall,30,66f,91 Stozek,Wlodzimierz,27,35ff,47f,65 Strauss,AdmiralLewis,210,221,223,237f,243 "super,"the,seehydrogenbomb Sylvester,JamesJ.,264 Szilard,Leo,294 T Tamarkin,JacobDavid,94,99f,116 Tarski,Alfred,29f,40,114,119,122 Taschek,Richard,144 Taylor,SirGeoffrey,144 Taylor,Theodore,221,256 Teller,Edward,146ff,162,164,169,172,184ff,193,202,209ff,219ff,223,231,235ff,239,251,253,257 TestBanTreaty,251 thermonuclearreactions,149,152,211ff,219ff; seealsohydrogenbomb thinkingprocess,22,178,180,184,289 Thomson,SirJosephJohn,60 Thucydides,102,244 Toll,John,220 Townes,Charles,255 Truman,PresidentHarryS.,210f,253 Tuck,James,228f,259 Turkevitch,Anthony,210 TwiningCommittee,256 U UFO(unidentifiedflyingobjects),301f Uhlenbeck,GeorgeE.,269 Ulam,Adam(brother),113ff,118,191

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Ulam,Andrzej(cousin),72,115 Ulam,Anna(mother),9,15,113 Ulam,Claire(daughter),161,174,230,235,239,244 Ulam,Françoise(Aron),44,125,133,145,161,174ff,178,189,190,215,230,239,244,260,262,269,270,272 Ulam,Jozef(father),9ff,18ff,113f Ulam,Michael(uncle),56f,108 Ulam,S.M. birth,9 brainillness,174ff inCambridge,England,58ff Colorado,Universityof,266ff doctor'sdegree,47 LosAlamos,141­171,188­208,209­224,249­265 marriage,125 mathematicalcuriosity,early,20ff "self­portrait,"271f U.S.citizenship,137f WorldWarII,outbreakof,116 Ulam,Szymon(uncle),16,114 UniversityofColorado,91,161,260 UniversityofSouthernCalifornia(USC),173f,179,186 UniversityofWisconsin,62,120,123,125 V Vassilief,Alexander,124 Veblen,Oswald,73f,115,283 Veblen,Thorstein,74 Verne,Jules,5,23,255,229 Vienna,10f,56f,261 Virgil,111 Vitali,G.,92 Voltaire,105,262 VonNeumann,John,seeNeumann,Johnvon W Walden,William,201 Wang,Hao,285 Watson,JamesD.,241,260,300 Wavre,Roland,45f Weierstrass,Karl,55,282 Weil,André,133 Weisskopf,VictorF.,152,162,168f,172,191,260ff,270 Weizsäcker,CarlFriedrichvon,216 Wells,H.G.,5,23 Wells,Mark,201 Weyl,Hermann(1885­1955),30,72,121,291 Wheeler,John,215,220f Whitehead,AlfredNorth(1861­1947),29,86,118f,138,285 Whitehead,Mrs.A.N.,30,118 Whitney,Hassler,91 Whyburn,GordonT.,107 Wiener,Norbert(1894­1964),45,91ff,109,137,160,294 Wiesner,Jerome,253ff,257 Wigner,Eugene,112,124,160,277 Woodward,RobertB.,86 Wundheiler,Alexander,119 Y Yang,ChenNing,294 Young,LawrenceC.,62 Z Zabuski,Norman,J.,228 Zawirski,Zygmund,21,23,65 Zygmund,Antoni,38,93,96,270

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