John von Neumann

Characterlife

JohnvonNeumann,afamousHungarian-AmericanmathematicianandcomputerScientists,physicistsandchemists.BorninaJewishfamilyinBudapest,HungaryonDecember28,1903.

VonNeumann’sfatherMaxwasyoungandvirtuousandpersonable.Withhisdiligence,witandgoodmanagement,hewasalreadyamongthebankersinBudapestwhenhewasyoung.VonNeumann'smotherwasakind-heartedwoman,virtuous,meek,andwell-educated.

VonNeumannhasshowngeniusinmathematicsandmemorysincehewasachild.Fromhischildhood,vonNeumannhasanunforgettabletalent.Attheageofsix,hewasabletospeakGreekThefatherwasjokingwitheachother.Whenhewassixyearsold,hewasabletodoeight-digitdivisionmentally.Whenhewaseightyearsold,hemasteredcalculus.Whenhewastenyearsold,hespentmonthsreadinga48-volumehistoryoftheworld.Theincidentwascompared,andthemilitarytheoriesandpoliticalstrategiesofthetwowerediscussed.Attheageoftwelve,heunderstoodandcomprehendedtheessentialsofBoller'smasterpiece"TheoryofFunctions".

Theessenceofcalculusisthemathematicalanalysisofinfinitesimalquantities.Mankindhasbeenexploringfiniteness,infinityandtherelationshipbetweenthemforalongtime.ThecalculusdiscoveredbyNewtonLeibnizinthe17thcenturyisanexcitingandgreatachievementinhumanexplorationofinfinity.Forthreehundredyears,ithasbeentheteachingcontentofhighereducationinstitutions.Withthedevelopmentofthetimes,calculusisconstantlychangingitsform,theconcepthasbecomemoreprecise,thebasictheoryissolid,andthereareevenmanyconciseandappropriatestatements.Butinanycase,itisstillrareforaneight-year-oldchildtounderstandcalculus.Althoughtheabove-mentionedrumorsarenotcompletelycredible,vonNeumann’sextraordinaryintelligenceistheconsensusofthepeopleheknew.

Inthesummerof1914,Johnenteredtheuniversitypreparatoryclass.OnJuly28ofthatyear,theAustro-HungarianEmpiredeclaredwaronSerbiaunderthepretextofopeningthepreludetotheFirstWorldWar.Duetocontinuouswarsandturmoil,vonNeumannandhisfamilyleftHungarybeforereturningtoBudapest.Ofcoursehisstudieswillalsobeaffected.Butinthegraduationexamination,vonNeumann'sresultswerestillamongthebest(exceptforsportsandwriting,allareA).

In1921,whenvonNeumannpassedthe"mature"exam,hewasalreadyregardedasamathematician.Hisfirstthesiswasco-writtenwithFickert,whenhewaslessthan18yearsold.Becauseofeconomicreasons,Maxaskedpeopletodissuadethe17-year-oldvonNeumannfromspecializinginmathematics.Later,thefatherandsonreachedanagreementandvonNeumannwenttostudychemistry.

Forthenextfouryears,vonNeumannregisteredasamathematicsstudentatBudapestUniversity,buthedidnotattendclasses.HejusttooktheexamontimeeveryyearandgotanAfortheexam.Atthesametime,vonNeumannenteredtheUniversityofBerlin(1921),andin1923enteredtheSwissFederalUniversityofTechnologyinZurichtostudychemistry.In1926,heobtainedauniversitydegreeinchemistryfromtheFederalUniversityofTechnologyinZurich.ByreturningtoBudapestUniversitytopassthecourseexaminationsattheendofeachsemester,healsoobtainedaPhDinmathematicsfromBudapestUniversity.

VonNeumann’swayofstudyingwithoutattendingclassesandonlytakingexamswasveryspecialatthetime,anditwascompletelyirregularinEuropeasawhole.ButthisirregularlearningmethodisverysuitableforvonNeumann.

DuringhisstayinZurich,vonNeumannoftenusedhisfreetimetostudymathematics,writearticles,andcommunicatewithmathematicians.DuringthisperiodvonNeumannwasinfluencedbytheideasofHilbertandhisstudentsSchmidtandWeylandbegantostudymathematicallogic.WeylandBoiawerealsoinZurichatthetime,andhehadarelationshipwiththem.OnceWeilleftZurichforashortperiodoftime,vonNeumannalsoattendedclassesonhisbehalf.Ingenuitycoupledwithuniquecultivation,vonNeumannisgrowingvigorously.Whenheendedhisschooldays,hehadalreadywanderedatsomeofthefrontiersinthethreefieldsofmathematics,physics,andchemistry.

Inthespringof1926,vonNeumannwenttotheUniversityofGöttingenasHilbert'sassistant.From1927to1929,vonNeumannworkedasapart-timelecturerattheUniversityofBerlin,duringwhichtimehepublishedarticlesonsettheory,algebraandquantumtheory.In1927,vonNeumannwenttoLviv,PolandtoattendtheMathematicsConference.Atthattime,hewasalreadywell-knownforhisworkonthefoundationsofmathematicsandsettheory.

In1929,vonNeumannwastransferredtoapart-timelecturerattheUniversityofHamburg.HewenttotheUnitedStatesforthefirsttimein1930andbecameaguestlectureratPrincetonUniversity.TheUnitedStates,whichisgoodatgatheringtalents,soonhiredvonNeumannasaguestprofessor.

VonNeumannoncecalculatedthattherearefewexistingandexpectedvacanciesinGermanuniversities.Accordingtohistypicalreasoning,thenumberofprofessorappointmentsthatcanbeobtainedinthreeyearsisthree.Thereareasmanyas40lecturersparticipatinginthecompetition.InPrinceton,vonNeumannreturnedtoEuropeeverysummer,until1933asaprofessoratthePrincetonInstituteofAdvancedStudies.Atthattime,theAdvancedResearchInstituteemployedsixprofessors,includingEinstein,andvonNeumann,whowasonly30yearsold,wastheyoungestofthem.

WhentheInstituteofAdvancedStudieswasfounded,Europeanvisitorswouldfindthattherewasanexcellentinformalandstrongresearchatmosphere.Theprofessors'officesaresetupintheuniversity's"beautifulbuilding",wherelifeisstable,thinkingisactive,andhigh-qualityresearchresultsareendless.Itcanbesaidthatthereisthelargestnumberofmathematicsandphysicsmindsinhistory.

In1930,vonNeumannandMaridaCorvezmarried.TheirdaughterMarinawasborninPrincetonin1935.VonNeumann’shouseoftenhostslong-lastingsocialgatherings,whichiswellknownfromfarandnear.VonNeumanndivorcedhiswifein1937,marriedClaraDanin1938,andreturnedtoPrincetontogether.DanstudiedmathematicswithvonNeumannandlaterbecameanexcellentprogrammer.AftermarryingClara,vonNeumann'shomeisstillameetingplaceforscientists,anditisstillsohospitable,whereeveryonewillfeelasmartatmosphere.

AftertheoutbreakoftheSecondWorldWarinEurope,vonNeumann'sactivitiessurpassedPrincetonandparticipatedinanumberofscientificresearchprojectsrelatedtotheanti-fascistwar.Since1943,hehasbeenaconsultantforthemanufactureofatomicbombs,andafterthewarhestillservedinmanygovernmentdepartmentsandcommittees.In1954,hebecameamemberoftheAmericanAtomicEnergyCommission.

VonNeumann’slong-timefriend,Strauss,ChairmanoftheAtomicEnergyCommission,oncecommentedonhim:Fromhisappointmenttothelateautumnof1955,vonNeumanndidagreatjobPretty.Hehasanunmatchedability.Themostdifficultproblemswillbebrokendownintoseeminglysimplethingsinhishands.Inthisway,hehasgreatlypromotedtheworkoftheAtomicEnergyCommission.

VonNeumann'shealthhasalwaysbeenverygood,butduetohisbusywork,hebegantofeelverytiredin1954.

Inthesummerof1955,X-raysrevealedthathewassufferingfromcancer,buthewasstillworkingandhisconditionexpanded.Later,hewasplacedinawheelchairandcontinuedtothink,givespeeches,andparticipateinmeetings.Long-termandrelentlessillnesstorturedhimandslowlystoppedallhisactivities.InApril1956,heenteredtheWalterReidHospitalinWashington,anddiedinthehospitalonFebruary8,1957attheageof53.

PersonalAchievements

VonNeumannisoneofthemostimportantmathematiciansofthe20thcentury.Hehasmadeoutstandingcontributionsinpureandappliedmathematics.Hisworkcanberoughlydividedintotwoperiods:before1940,itwasmainlypurelymathematicalresearch:asimpleandcleartheoryofordinalnumberswasputforwardinmathematicallogic,andanewaxiomatizationofsettheorywasmade,whichclearlydistinguishedbetweensetsandclasses.Later,hestudiedthespectrumtheoryoflinearself-adjointoperatorsonHilbertspace,thuslayingamathematicalfoundationforquantummechanics;since1930,heprovedthattheaverageergodictheoremopenedupanewfieldofergodictheory;in1933,heusedThecompactgroupsolvedHilbert'sfifthproblem;inaddition,healsomadegroundbreakingcontributionsinmeasurementtheory,latticetheory,andcontinuousgeometry;from1936to1943,heandMurraycollaboratedtocreateoperatorsRingtheory,theso-calledvonNeumannalgebra.

After1940,vonNeumannturnedtoappliedmathematics.Ifhispuremathematicalachievementsbelongtothefieldofmathematics,thenhisworkinmechanics,economics,numericalanalysis,andelectroniccomputersbelongstoallmankind.AtthebeginningofWorldWarII,vonNeumannstudiedthemotionofcompressiblegasduetotheneedsofwarfare,establishedshockwavetheoryandturbulencetheory,anddevelopedfluidmechanics.Since1942,hehascollaboratedwithMogensterntowrite"Thebook"GameTheoryandEconomicBehavior",whichisaclassicworkingametheory(alsoknownasgametheory),madehimoneofthefoundersofmathematicaleconomics.

VonNeumannmadesuggestionsforthedesignoftheworld’sfirstelectroniccomputerENIAC(ElectronicDigitalIntegralComputer).InMarch1945,hedraftedabrandnew"Storedprogramgeneralelectroniccomputerprogram"-EDVAC(abbreviationforElectronicDiscreteVariableAutomaticComputer).Thishasadecisiveinfluenceonthedesignoflatercomputers,especiallythedeterminationofthecomputer'sstructure,theuseofstoredprogramsandbinarycodes,etc.,whicharestillfollowedbyelectroniccomputerdesigners.

In1946,vonNeumannbegantostudyprogrammingproblems.Hewasoneofthecreatorsofmodernnumericalanalysis-computationalmathematics.Hefirststudiedlinearalgebraandarithmeticnumericalcalculations,andlaterfocusedonDiscretizationandstabilityofnonlineardifferentialequations,andgivesanestimateoftheerror.Heassistedinthedevelopmentofsomealgorithms,especiallytheMonteCarlomethod.

Inthelate1940s,hebegantostudyautomatatheory,generallogictheoryandself-replicatingsystems.Inthelastmomentsofhislife,hedeeplycomparednaturalautomatawithartificialautomata.Hisunfinishedmanuscriptafterhisdeathwaspublishedin1958underthename"ComputerandHumanBrain".

ThemainworksofvonNeumannarecollectedin"TheCompleteWorksofVonNeumann"(6volumes,1961).

Whetherinpuremathematicsorinappliedmathematicsresearch,vonNeumannhasshownoutstandingtalentsandachievedmanyfar-reachingmajorresults.Constantlychangingresearchtopics,andoftenachievingachievementsinthecross-penetrationofseveraldisciplinesarehischaracteristics.

Simplyspeaking,hisessencecontributionistwopoints:binarythinkingandprogrammemorythinking.

Whenlookingbackonthebrilliantdevelopmentofscienceandtechnologyinthe20thcentury,oneofthemostoutstandingmathematiciansofthe20thcentury,vonNeumann,mustbementioned.Asweallknow,theelectroniccomputerinventedin1946greatlypromotedtheprogressofscienceandtechnology,andgreatlypromotedtheprogressofsociallife.InviewofthekeyrolethatvonNeumannplayedintheinventionoftheelectroniccomputer,hewashailedasthe"fatherofcomputers"byWesterners.Ineconomics,healsohasbreakthroughachievementsandisknownasthe"fatherofgametheory."Inthefieldofphysics,"TheMathematicalBasisofQuantumMechanics"writtenbyvonNeumanninthe1930shasbeenprovedtobeextremelyvaluabletothedevelopmentofatomicphysics.Healsohasconsiderableattainmentsinchemistry,havingobtainedauniversitydegreefromtheDepartmentofChemistryoftheZurichInstituteofTechnology.LikeHayek,whoisalsoaJew,heisworthyofbeingoneofthegreatestall-roundersofthelastcentury.

VonNeumannhasdonepioneeringworkinmanyfieldsofmathematicsandmadesignificantcontributions.BeforetheSecondWorldWar,hewasmainlyengagedintheresearchofoperatortheoryandsettheory.The1923paperonover-limitordinalnumbersinsettheoryshowedvonNeumann'suniquewayandstyleindealingwithsettheoryproblems.Heaxiomatizedsettheory,andhisaxiomaticsystemlaidthefoundationforaxiomaticsettheory.Startingfromaxioms,heusedalgebraicmethodstoderivemanyimportantconcepts,basicoperations,andimportanttheoremsinsettheory.Especiallyinapaperin1925,vonNeumannpointedoutthatthereareundecidablepropositionsinanyaxiomaticsystem.

In1933,vonNeumannsolvedHilbert'sfifthproblem,whichprovedthatthelocalEuclideancompactgroupisaLiegroup.In1934,heunifiedthetheoryofcompactgroupswithBohr'stheoryofalmostperiodicfunctions.Healsohasadeepunderstandingofthestructureofgeneraltopologicalgroups,andhasfiguredoutthatitsalgebraicstructureandtopologicalstructureareconsistentwithrealnumbers.Hedidpioneeringworkonoperatoralgebraandlaiditstheoreticalfoundation,thusestablishingthenewbranchofmathematics,operatoralgebra.ThisbranchiscalledvonNeumannalgebraincontemporaryrelatedmathematicsliterature.Thisisanaturalgeneralizationofmatrixalgebrainafinite-dimensionalspace.VonNeumannalsofoundedgametheory,anotherimportantbranchofmodernmathematics.In1944,hepublishedafoundationalandimportantpaper"GameTheoryandEconomicBehavior".Thepapercontainsanexplanationofthepurelymathematicalformofgametheoryandadetailedexplanationoftheactualgameapplication.Thearticlealsocontainsteachingideassuchasstatisticaltheory.VonNeumannhasdoneimportantworkinthefieldsoflatticetheory,continuumgeometry,theoreticalphysics,dynamics,continuummechanics,meteorologicalcalculation,atomicenergyandeconomics.

VonNeumann’sgreatestcontributiontomankindishispioneeringworkoncomputerscience,computertechnology,numericalanalysis,andgametheoryineconomics.

ItisgenerallybelievedthattheENIACmachineistheworld'sfirstelectroniccomputer.ItwasdevelopedbyAmericanscientistsandstartedoperationinPhiladelphiaonFebruary14,1946.Infact,the"Korosas"computerdevelopedbyBritishscientistssuchasTommyandFerrolswasintroducedmorethantwoyearsearlierthantheENIACmachine,anditstartedoperatinginBletchleyParkonJanuary10,1944.TheENIACmachineprovesthattheelectronicvacuumtechnologycangreatlyimprovethecalculationtechnology.However,theENIACmachineitselfhastwomajorshortcomings:(1)Nomemory;(2)Itusesawiringboardforcontrol,anditeventakesafewdaystoconnect,andthecalculationspeedisalsoItwasoffsetbythiswork.MokleyandEckertoftheENIACmachinedevelopmentteamobviouslyfeltthis,andtheyalsowantedtostartdevelopinganothercomputerassoonaspossibletoimproveit.

In1944,Neumannparticipatedinthedevelopmentoftheatomicbomb,whichinvolvedextremelydifficultcalculations.Inthestudyofthenuclearreactionprocess,itisnecessarytomakea"yes"or"no"answertothepropagationofareaction.Solvingthisproblemusuallyrequiresbillionsofmathematicaloperationsandlogicalinstructions.Althoughthefinaldataisnotrequiredtobeveryaccurate,allintermediateoperationsareindispensableandmustbeasaccurateaspossible.TheLosAlamoslaboratorywhereheisemployedemploysmorethanonehundredfemalecomputingstaffforthispurpose,usingdesktopcomputerstocalculatefrommorningtonight,whichisstillfarfromsatisfyingtheneeds.Endlessnumbersandlogicalinstructionsexhausthumanwisdomandenergylikeadesert.

Neumann,whowastroubledbycomputers,learnedoftheENIACcomputerdevelopmentplanbyaveryaccidentalopportunity.Fromthenon,hedevotedhimselftothemagnificentbusinessofcomputerdevelopmentandestablishedthegreatestachievementinhislife..

Onedayinthesummerof1944,Neumann,whowaswaitingatthetrainstation,metGoldsteinbychanceandhadabriefconversationwithhim.Atthattime,GoldsteinwasthemilitarychiefoftheUSBallisticsLaboratory,andhewasparticipatinginthedevelopmentoftheENIACcomputer.Duringtheconversation,GoldsteintoldNeumannaboutthedevelopmentofENIAC.ThevisionaryNeumannwasattractedbythisdevelopmentplan,andherealizedthefar-reachingsignificanceofthiswork.

VonNeumannwasintroducedbyLieutenantGoldsteinoftheENIACmachinedevelopmentteamafterjoiningtheENIACmachinedevelopmentteam,andledthisgroupofinnovativeyoungscientificandtechnologicalpersonneltomarchtowardsahighergoal.In1945,onthebasisofjointdiscussions,theypublishedanew"storedprogramgeneralelectroniccomputerprogram"-EDVAC(abbreviationforElectronicDiscreteVariableAutomaticComputer).Duringthisprocess,vonNeumannshowedhisstrongbasicknowledgeofmathematicsandscience,andgavefullplaytohisadvisoryroleandabilitytoexploreproblemsandcomprehensiveanalysis.Neumanndrafteda101-pagesummaryreportunderthetitleof"DraftReportonEDVAC".Thereportextensivelyandconcretelyintroducedthenewideasofmanufacturingelectroniccomputersandprogramming.Thisreportisanepoch-makingdocumentinthehistoryofcomputerdevelopment.Itdeclarestotheworldthattheageofelectroniccomputershasbegun.

TheEDVACprogramclearlyestablishedthatthenewmachineiscomposedoffiveparts,including:arithmeticunit,controller,memory,inputandoutputequipment,anddescribedthefunctionsandrelationshipsofthesefiveparts.Inthereport,NeumannfurtherdemonstratedthetwomajordesignideasinEDVAC,settingamilestoneforcomputerdesign.

Oneofthedesignideasisbinary.Accordingtothecharacteristicsofthebistableoperationofelectroniccomponents,herecommendstheuseofbinaryinelectroniccomputers.Thereportmentionedtheadvantagesofbinary,andpredictedthattheuseofbinarywillgreatlysimplifythelogiccircuitofthemachine.

Thebasicworkingprincipleofacomputeristostoreprogramsandprogramcontrol,whichwasproposedbytheworld-renownedmathematicianvonNeumann.TheAmericanHungarianmathematicianvonNeumannisknownasthe"fatherofcomputers."

PracticehasprovedthecorrectnessofNeumann'sprediction.Nowadays,theapplicationoflogicalgebrahasbecomeanimportantmeansofdesigningelectroniccomputers,andthemainlogiccircuitsusedinEDVAChavebeenusedallthetime,buttheengineeringmethodsandanalysismethodsofrealizinglogiccircuitshavebeenimproved.

Classicaltheory

VonNeumannArchitecture

Whenitcomestothedevelopmentofcomputers,youcannotfailtomentiontheAmericanscientistvonNeumann.Sincethebeginningofthe20thcentury,physicsandelectronicsscientistshavebeenarguingaboutwhatstructureshouldbeusedtomakeamachinethatcanperformnumericalcalculations.Peoplearetroubledbythedecimalnumber,acountingmethodthathumansareusedto.Therefore,thecallforthedevelopmentofanalogcomputersatthattimewasevenlouderandmorepowerful.Inthemid-1930s,theAmericanscientistvonNeumannboldlyproposedtoabandonthedecimalsystemandadoptthebinarysystemasthebasisofthenumbersystemfordigitalcomputers.Atthesametime,healsosaidthatthecalculationprogramispre-programmed,andthenthecomputerperformsthenumericalcalculationworkinaccordancewiththecalculationsequenceestablishedbypeopleinadvance.

ThemainpointofvonNeumann'stheoryisthatthenumbersystemofdigitalcomputersisbinary;thecomputershouldexecuteintheorderofprograms.

PeoplecallthisvonNeumanntheorythevonNeumannarchitecture.FromENIAC(ENIACisnotthevonNeumannsystem)tothecurrentmostadvancedcomputersallusethevonNeumannarchitecture.SovonNeumannisthefatherofdigitalcomputers.

ThecomputerconstructedaccordingtothevonNeumannarchitecturemusthavethefollowingfunctions:

Sendtherequiredprogramsanddatatothecomputer.

Musthavetheabilitytolong-termmemoryprocedures,data,intermediateresultsandfinalcalculationresults.

Theabilitytocompleteavarietyofarithmetic,logicaloperationsanddatatransmissionandotherdataprocessingcapabilities.

Itcancontroltheprogramdirectionaccordingtotheneeds,andcancontrolthecoordinatedoperationofthevariouspartsofthemachineaccordingtotheinstructions.

Abletooutputtheprocessingresultstotheuserasrequired.

Inordertocompletetheabove-mentionedfunctions,thecomputermusthavefivebasiccomponents,including:

Inputdevicesforinputtingdataandprograms

Memoryforstoringprogramsanddata

Computerthatcompletesdataprocessing

Controllerthatcontrolsprogramexecution

Outputdevicethatoutputsprocessingresults

Theoreticalbasis

VonNeumann’sfirstpaperwasco-writtenwithFichter.ItwasabouttheextensionoftheFyettheoremoftheChebyshevpolynomialrootmethod.Thedateindicatedwas1922,atthattimeVonNeumannisnotyet18yearsold.Anotherarticlediscussesaconsistentdensesequenceofnumbers,writteninHungarian.TheselectionofthetitleandthesimplicityoftheproofmethodrevealtheintuitivecombinationofvonNeumann'salgebraicskillsandsettheory.

In1923,whenvonNeumannwasauniversitystudentinZurich,hepublishedapaperonover-limitordinalnumbers.Thefirstsentenceofthearticlebluntlystatedthat"thepurposeofthisarticleistoconcretizeandrefineCantor'sconceptofordinalnumbers."Hisdefinitionofordinalnumbershasbeenwidelyadopted.

ItisvonNeumann’sdesiretostronglyseektodiscussaxiomatization.From1925to1929,mostofhisarticlestriedtoimplementthespiritofaxiomatization,evenintheoreticalphysicsresearch.Sotoo.Atthattime,histreatmentofsettheorywasparticularlynotformalized.Inhis1925doctoraldissertationontheaxiomsystemofsettheory,hesaidatthebeginning,"Thepurposeofthisarticleistologicallyaxiomatizesettheory.Discuss".

Interestingly,vonNeumanninhisthesisforeseethelimitationsofanyformofaxiomsystem,whichvaguelyremindspeopleoftheincompletenesstheoremlaterprovedbyGödel.Forthisarticle,ProfessorFrankel,awell-knownlogicianandoneofthefoundersoftheaxiomsettheory,oncecommented:"IcannotinsistthatIhaveunderstoodeverything(ofthearticle),butIcansayitwithconfidence.Itisanoutstandingjob,andagiantcanbeseenthroughhim."

In1928,vonNeumannpublishedthepaper"TheAxiomatizationofSetTheory",whichwasanaxiomatictreatmentoftheabove-mentionedsettheory.Thesystemisveryconcise.Itusesthefirsttypeobjectandthesecondtypeobjecttorepresentthesetandthenatureofthesetinthenaivesettheory.Theaxiomsofthesystemcanbewritteninalittlemorethanonepage.ItisenoughtoestablishthenaiveAllthecontentofsettheory,andtoestablishtheentiremodernmathematics.

VonNeumann'ssystemmaybethefirstbasisforsettheory.Thefiniteaxiomsusedhavealogicalstructureassimpleaselementarygeometry.Startingfromtheaxioms,VonNeumann'sabilitytousealgebraicmethodstoderivemanyimportantconceptsinsettheoryissimplyamazing.Allthesehavealsopreparedtheconditionsforhisfutureinterestincomputersand"mechanized"proofs.

Inthelate1920s,vonNeumannparticipatedinHilbert'smeta-mathematicsprojectandpublishedseveralpapersprovingthatsomearithmeticaxiomswerenotcontradictory.The1927paper"OnHilbert'sProofTheory"isthemosteye-catching.Itsthemeistodiscusshowtofreemathematicsfromcontradictions.ThearticleemphasizedthatthisquestionraisedanddevelopedbyHilbertetal.wasverycomplicatedandhadnotbeensatisfactorilyansweredatthattime.ItalsopointsoutthatAckerman'sproofofexcludingcontradictionscannotberealizedinclassicalanalysis.Forthisreason,vonNeumannmadeastrictproofoffinitenessforacertainsubsystem.ThisseemsnotfarfromthefinalanswerHilbertisseeking.Atthistime,Gödelprovedtheincompletenesstheoremin1930.Thetheoremassertsthatinanon-contradictoryformalsystemincludingelementaryarithmetic(orsettheory),thenon-contradictionofthesystemcannotbeprovedinthesystem.Sofar,vonNeumanncanonlysuspendresearchinthisarea.

VonNeumannalsogotspecialresultsaboutsettheoryitself.Hisinterestinmathematicalfoundationsandsettheorycontinueduntiltheendofhislife.

PureMathematics

From1930to1940,vonNeumann'sachievementsinpuremathematicsweremoreconcentrated,hiscreationbecamemoremature,andhisreputationincreased.Later,inaquestion-and-answerformfortheNationalAcademyofSciences,vonNeumannchosethemathematicalfoundationofquantumtheory,thetheoryofoperatorrings,andtheergodictheoremashismostimportantmathematicalwork.

In1927,vonNeumannwasalreadyengagedinresearchinthefieldofquantummechanics.HeandHilbertandNordamjointlypublishedthepaper"TheFundamentalsofQuantumMechanics".ThebasisofthisarticleisHilbert'slectureonthenewdevelopmentofquantummechanicsinthewinterof1926.Nordamhelpedpreparethelecture,andvonNeumannworkedonthemathematicalformalizationofthesubject.Thepurposeofthearticleistoreplacetheexactfunctionalrelationsinclassicalmechanicswithprobabilityrelations.Hilbert'smeta-mathematicsandaxiomaticsolutionshavebeenappliedinthisdynamicfield,andtheisomorphicrelationshipbetweentheoreticalphysicsandthecorrespondingmathematicalsystemhasbeenobtained.Inanycase,thehistoricalimportanceandinfluenceofthisarticlewillnotbetoohigh.Inhisarticle,VonNeumannalsodiscussedtheoutlineoftheoperationsofobservableoperatorsinphysicsandthepropertiesofHermiteoperators.Undoubtedly,thesecontentsconstitutethepreludetothebook"TheMathematicalBasisofQuantumMechanics".

In1932,theworld-famousSpringerPublishingHousepublishedhis"TheMathematicalFundamentalsofQuantumMechanics".ItisoneofvonNeumann’smainworks.ThefirsteditionisinGerman.Itwaspublishedin1943.TheFrenchversionwaspublishedinSpanishin1949andtranslatedintoEnglishin1955.Itisstillaclassicinthisregard.Ofcourse,hehasalsodonealotofimportantworkinquantumstatistics,quantumthermodynamics,andgravitationalfields.

Objectivelyspeaking,inthehistoryofquantummechanics,vonNeumannhasmadeatleasttwoimportantcontributions:Dirac’smathematicaltreatmentofquantumtheoryisnotrigorousenoughinacertainsense.NeumanndevelopedtheHilbertoperatortheorythroughthestudyofunboundedoperatorstomakeupforthisdeficiency.Inaddition,vonNeumannclearlypointedoutthatthestatisticalcharacteristicsofquantumtheoryarenotduetothestateoftheobserverengagedinthemeasurement.Causedbyunknown.WiththehelpofHilbertspaceoperatortheory,heprovedthatalltheassumptionsofquantumtheoryincludingtheassociativityofgeneralphysicalquantitieswillinevitablyleadtothisresult.

ForthecontributionofvonNeumann,NobelPrizewinnerWigeneroncemadethefollowingevaluation:"ThecontributiontoquantummechanicsistoensurethatheisinthefieldofcontemporaryphysicsSpecialstatus."

InvonNeumann’swork,thespectrumtheoryandringtheoryofoperatorsonHilbertspaceoccupiesanimportantdominantposition.ArticlesinthisareaareaboutItaccountsforone-thirdofhispublishedpapers.Theyincludeextremelydetailedanalysisofthepropertiesoflinearoperators,andalgebraicresearchonoperatorringsininfinitedimensionalspaces.

Thetheoryofoperatorringsbeganinthesecondhalfof1930.VonNeumannwasveryfamiliarwiththenon-commutativealgebraofNottandArden,andsoonuseditforboundedlinesinHilbertspace.Thealgebracomposedofsexoperatorswentup,andlatergenerationscalleditvonNeumannoperatoralgebra.

From1936to1940,vonNeumannpublishedsixpapersonnon-commutativeoperatorrings,whichcanbedescribedasamasterpieceofanalysisinthe20thcentury,anditsinfluencehasbeenextendedtothisday.VonNeumannoncesaidin"TheMathematicalBasisofQuantumMechanics":TheearliestideasproposedbyHilbertcanprovideanappropriatefoundationforthequantumtheoryofphysics,withouttheneedtointroducenewphysicaltheories.Mathematicalconception.Hisresearchresultsonoperatorringsfulfilledthisgoal.VonNeumann'sinterestinthissubjecthasbeenthroughouthisentirecareer.

AnamazinggrowthpointinthetheoryofoperatorringsisthecontinuousgeometrynamedbyvonNeumann.Thedimensionsofordinarygeometryareintegers1,2,3,etc.VonNeumannhasseeninhisworksthatwhatdeterminesthedimensionalstructureofaspaceisactuallytherotationgroupitallows.Therefore,thedimensionalitycannolongerbeaninteger,andthegeometryofthecontinuousseriesspaceisfinallyproposed.

In1932,vonNeumannpublishedapaperontheergodictheory,whichsolvedtheproofoftheergodictheorem,andexpresseditwiththeoperatortheory,whichisthewholeofthestrictprocessingoftheergodichypothesisinstatisticalmechanicsThefirstprecisemathematicalresultobtainedintheresearchfield.ThisachievementofvonNeumannmaybeattributedagaintohismasteryofmathematicalanalysismethodsinfluencedbysettheoryandthemethodshecreatedinthestudyofHilbertoperators.Itisoneofthemostinfluentialachievementsinthefieldofmathematicalanalysisinthe20thcentury,anditalsomarksthebeginningofageneralresearchinthefieldofmathematicalphysicsthatisapproachingaccuratemodernanalysis.

Inaddition,vonNeumannhasalsomademanyachievementsinrealvariablefunctiontheory,measuretheory,topology,continuum,latticetheoryandothermathematicalfields.Inthatfamousspeechin1900,Hilbertputforward23questionsfor20thcenturymathematicsresearch.VonNeumannalsomadeadecisivecontributiontosolvingHilbert'sfifthproblem.

AppliedMathematics

1940wasaturningpointinvonNeumann’sscientificcareer.Beforethat,hewasapuremathematicianwhowasacquaintedwithphysics;afterthat,hebecameasuperbappliedmathematicianwithafirmgraspofpuremathematics.Hebegantopayattentiontopartialdifferentialequations,themostimportanttoolforapplyingmathematicstothefieldofphysicsatthattime.Atthesametime,hecontinuedtoinnovateandappliednon-classicalmathematicstotwonewfields:gametheoryandelectroniccomputers.

VonNeumann'stransformationcamefromhislong-termloveformathematicalandphysicsproblemsontheonehand;ontheotherhand,itcamefromtheneedsofsocietyatthattime.AftertheoutbreakoftheSecondWorldWar,vonNeumannwascalledtoparticipateinmanymilitaryscientificresearchprojectsandengineeringprojects.From1940to1957,heservedasthescientificconsultantoftheAberdeenTestBallisticResearchLaboratoryinMaryland;from1941to1955intheWashingtonNavalOrdnanceBureau;from1943to1955astheconsultantoftheLosAlamosLaboratory;from1950to1955,theArmySpecialWeaponsDesignCommitteemember;1951～1957.MemberoftheWashingtonScientificAdvisoryBoardoftheUSAirForce;MemberoftheAtomicEnergyTechnologyAdvisoryGroupfrom1953to1957;ChairmanoftheMissileAdvisoryCommitteefrom1954to1957.

VonNeumannhasstudiedcontinuummechanics.Hehasbeeninterestedinturbulencephenomenaforalongtime.In1937,hepaidattentiontothediscussionofthepossibilityofstatisticalprocessingoftheNavier-Stokesequations(Englishname:Navier-Stokesequations).In1949,hewrote"TheLatestTheoryofTurbulence"fortheNavalResearchDepartment.

VonNeumannhasstudiedshockwaves.Mostofhisworkinthisfieldcomesdirectlyfromnationaldefenseneeds.Hiscontributiontotheinteractionofshockwavesisremarkable.OneoftheresultsisthathefirstrigorouslyprovedtheChapman-Rugerhypothesis,whichisrelatedtothecombustioncausedbyshockwaves.Thesystematicresearchonshockwavereflectiontheorystartedwithhis"ProgressReportonShockWaveTheory".

VonNeumannstudiedmeteorology.Forquitesometime,theextremelydifficultproblemsposedbythefluiddynamicsequationsoftheearth'satmosphericmotionhavebeenattractinghim.Withtheadventofelectroniccomputers,itispossibletoconductnumericalresearchandanalysisonthisproblem.VonNeumann'sfirsthighlyscaledcalculationdealtwithatwo-dimensionalmodel,whichwasrelatedtothegeostrophicapproximation.Hebelievesthatpeoplewilleventuallybeabletounderstand,calculateandachievecontroltochangetheclimate.

VonNeumannalsoproposedusingfusiontodetonatenuclearfuelandsupportedthedevelopmentofhydrogenbombs.In1947,thearmyissuedacommendationordercommendinghimasaphysicist,engineer,weapondesigner,andpatriot.

GameTheory

VonNeumannusedhistalentsnotonlyinweaponsresearch,butalsoinsocialresearch.In1928,vonNeumannprovedthebasicprinciplesofgametheory,thusproclaimingtheofficialbirthofgametheory.Thegametheorycreatedbyhimisundoubtedlyhismostenviableoutstandingachievementinappliedmathematics.Nowadays,gametheorymainlyreferstospecificmathematicalmethodsforstudyingsocialphenomena.Itsbasicideaistopayattentiontothesimilarityofbehaviorssuchasbargaining,negotiation,ganging,andprofitdistributionamongcompetitorsinindoorgamessuchaschessandpoker.

Someideasofgametheoryexistedintheearly1920s,andtherealcreationhadtostartwithvonNeumann’s1928thesisonsocialgametheory.Inthisarticle,heprovedtheminimum-maximumtheorem,whichisusedtodealwiththemostbasictwo-persongameproblem.Ifanyoneofthetwosidesofthestrategyconsidersthemaximumpossiblelossforeachpossiblestrategy,andchoosestheonewiththesmallest"maximumloss"asthe"optimal"strategy,thenfromastatisticalpointofview,heAbletoensurethattheprogramisthebest.Theworkinthisareahasgenerallybeencompleted.Inthesamepaper,vonNeumannalsoclearlystatedthegeneralcountermeasuresamongnplayers.

Gametheoryisalsousedineconomics.Mathematicalresearchmethodsineconomictheorycanberoughlydividedintopuretheorywithqualitativeresearchasthegoalandeconometricswithempiricalandstatisticalresearchasthegoal.Theformeriscalledmathematicaleconomics,anditwasformallyestablishedafterthe1940s.Nomatterinideologyormethod,itisobviouslyinfluencedbygametheory.

Mathematicaleconomics,inthepastimitatingthetechniquesofclassicalmathematicalphysics,themathematicaltoolsusedaremainlycalculusanddifferentialequations,andeconomicproblemsaretreatedasclassicalmechanicsproblems.Obviously,tradefairsattendedbydozensofbusinessmen,analyzedandprocessedbyclassicalmathematics,arefarmorecomplexthanthemotionsofplanetsinthesolarsystem.Theeffectofthismethodisoftendifficulttopredict.VonNeumannresolutelyabandonedthissimplemechanicalanalogyandreplaceditwithanovelgametheoryviewandnewmathematics—andconvexity.

In1944,"GameTheoryandEconomicBehavior"co-authoredbyvonNeumannandMorganSternwasthefoundationalworkinthisregard.Thetwo-persongameisextendedtothen-persongamestructureandthegametheorysystemisappliedtotheeconomicfield,thuslayingthefoundationandtheoreticalsystemofthisdiscipline.Thethesiscontainsanexplanationofthepurelymathematicalformofgametheoryandadetailedexplanationofpracticalapplications.Thispaperandthediscussionofsomebasicissuesofeconomictheorieshaveledtovariousresearchesoneconomicbehaviorandcertainsociologicalissues.Today,thisisawidelyusedandincreasinglyabundantsubject.Mathematicssubjects.Somescientistsenthusiasticallypraiseditas"oneofthegreatestscientificcontributionsofthefirsthalfofthe20thcentury."

Computers

ThelastsubjectthatcontributedtovonNeumann’sreputationwaselectroniccomputersandautomationtheory.

AsearlyasinLosAlamos,vonNeumannclearlysawthatevenifsometheoreticalphysicsresearchisjusttoobtainqualitativeresults,analyticalresearchaloneisnotenough,anditmustbesupplemented.Itiscalculatednumerically.Thetimeittakestoperformmanualcalculationsorusedesktopcomputersisintolerable,sovonNeumannstartedtoengageinresearchonelectroniccomputersandcalculationmethodswithgreatenthusiasm.

From1944to1945,vonNeumannformedthebasicmethodusedtodaytotransformasetofmathematicalprocessesintoacomputerinstructionlanguage.Atthattime,electroniccomputers(suchasENIAC)lackedflexibilityanduniversality.sex.VonNeumannmadeasignificantcontributiontoovercomingtheaboveshortcomingsonthefixedanduniversallinesysteminthemachine,ontheconceptof"flowgraph",andontheconceptof"code".Althoughtomathematicallogicians,thisarrangementisobvious.

ThedevelopmentofcomputerengineeringshouldalsobegreatlyattributedtovonNeumann.Thelogicschemeofcomputers,theselectionofstorage,speed,basicinstructions,andthedesignoftheinteractionbetweencircuitsinmoderncomputersarealldeeplyinfluencedbyvonNeumann'sthoughts.HenotonlyparticipatedinthedevelopmentoftheelectronictubecomponentcomputerENIAC,butalsopersonallysupervisedtheconstructionofacomputeratthePrincetonInstituteforAdvancedStudy.Earlier,vonNeumannandMoore'steamwroteabrand-newstorageprogramgeneralelectroniccomputerprogramEDVAC.The101-pagereportmadeasensationinthemathematicscommunity.ThePrincetonInstituteforAdvancedStudy,whichhasbeenspecializingintheoreticalresearch,alsoapprovedvonNeumanntobuildacomputer,basedonthisreport.

Electroniccomputersthataretensofmillionsoftimesfasterthanmanualcalculationshavenotonlygreatlypromotedtheprogressofnumericalanalysis,butalsostimulatedtheemergenceofnewmethodsinthebasicaspectsofmathematicalanalysisitself.Amongthem,thevigorousdevelopmentoftheMonteCarlomethodformulatedbyvonNeumannetal.,whichusesrandomnumberstodealwithdeterministicmathematicalproblems,isaprominentexample.

Theprecisemathematicalexpressionoftheprinciplesofmathematicalphysicsinthe19thcenturyseemstobeverylackinginmodernphysics.Thecomplexstructuresappearinginthestudyofelementaryparticlesaredazzling,andthereisstilllittlehopeoffindingacomprehensivemathematicaltheorysoon.Fromacomprehensivepointofview,withoutmentioningtheanalysisdifficultiesencounteredwhendealingwithsomepartialdifferentialequations,itisnothopefultoobtainaccuratesolutions.Alloftheseforcespeopletoseeknewmathematicalmodelsthatcanbeprocessedwiththehelpofelectroniccomputers.VonNeumanncontributedmanygeniusmethodstothis:mostofthemarecontainedinvariousexperimentalreports.Fromsolvingnumericalapproximatesolutionsofpartialdifferentialequations,tolong-termweathernumericalforecasting,andfinallyachievingclimatecontrol.

InthelastfewyearsofvonNeumann’slife,histhoughtswerestillveryactive.Heintegratedtheresultsofhisearlyresearchonlogicandhisworkoncomputers,andexpandedhishorizonstogeneralautomatatheory.Heattackedthemostcomplicatedproblemswithhisuniquecourage:howtouseunreliablecomponentstodesignreliableautomata,andbuildautomatathathecanreproduce.Fromthis,herealizedsomesimilaritiesbetweenthecomputerandthehumanbrain.TheresearchinthisareawasreflectedinSilleman'slecture;itwasonlyafterhisdeaththatsomeonepublishedaseparatebookunderthename"ComputerandtheHumanBrain."Althoughthisisanunfinishedwork,somequantitativeresultsobtainedfromhispreciseanalysisandcomparisonofhumanbrainsandcomputersystemsstilldonotloseitsimportantacademicvalue.

Publishedbooks

Mainworks

《ClassicalMechanicsOperatorMethod"

"TheMathematicalBasisofQuantumMechanics"(1932)

AfterthedeathofvonNeumann,theunfinishedmanuscriptwaspublishedin1958as"ComputerandHumanBrain"."Publishedunderthename.Hismainworksarecollectedinsixvolumesof"TheCompleteWorksofVonNeumann",whichwaspublishedin1961.

Inaddition,vonNeumann'sbook"GameTheoryandEconomicBehavior"(co-authoredwithMorganston),publishedinthe1940s,madehimamonumentinthefieldofeconomicsanddecisionscience.Heisrecognizedbyeconomistsasthefatherofgametheory.Atthattime,theyoungJohnNashbegantoresearchanddevelopthisfieldwhilehewasstudyinginPrinceton,andin1994hewontheNobelPrizeinEconomicsforhisoutstandingcontributionstogametheory.

"ProgramMemory"isanothermasterpieceofNeumann.ThroughtheinvestigationofENIAC,Neumannkeenlygraspeditsbiggestweakness-norealmemory.ENIAChasonly20temporaryregisters,anditsprogramsareplug-intype,andtheinstructionsarestoredinothercircuitsofthecomputer.Inthisway,beforesolvingtheproblem,youmustfirstthinkaboutalltheinstructionsyouneed,andconnectthecorrespondingcircuitsmanually.Thispreparationcantakehoursorevendays,andthecalculationitselfonlytakesafewminutes.Thereisabigcontradictionbetweenthehighspeedofcalculationandthemanualworkoftheprogram.

Inresponsetothisproblem,Neumannproposedtheideaof​​programmemory:storethecalculationprograminthememoryofthemachine,andtheprogrammeronlyneedstolookforthecalculationinstructionsinthememory,andthemachinewillcalculatebyitself.Inthisway,Thereisnoneedtoreprogrameveryproblem,whichgreatlyspeedsupthecalculationprocess.Thisideamarkstherealizationofautomaticcalculationsandthematurityofelectroniccomputers.Ithasbecomethebasicprincipleofelectroniccomputerdesign.

InJulyandAugust1946,vonNeumann,Goldstein,andBurksdevelopedtheIAScomputerfortheInstituteofAdvancedStudiesofPrincetonUniversityonthebasisoftheENIACplan,andproposedAmorecompletedesignreport"APreliminaryStudyontheLogicDesignofElectronicComputers."Theabovetwodocumentswithboththeoriesandspecificdesignscreateda"computerfever"forthefirsttimeintheworld.Theircomprehensivedesignideasarethefamous"VonNeumannmachine",whosecenterisstorage.Theprincipleoftheprogram-instructionsanddataarestoredtogether(memorymachine).Thisconceptishailedas"amilestoneinthehistoryofcomputerdevelopment."Itmarkstherealbeginningoftheelectroniccomputereraandguidesfuturecomputerdesign.Naturally,everythingisalwaysdeveloping.Withtheadvancementofscienceandtechnology,peopletodayrecognizetheshortcomingsofthe"vonNeumannmachine",whichhindersthefurtherimprovementofcomputerspeed,andputforwardthe"non-vonNeumannmachine".Theideaof​​"Emanji".

VonNeumannalsoactivelyparticipatedinthepromotionandapplicationofcomputers,andmadeoutstandingcontributionstohowtoprogramandperformnumericalcalculations.VonNeumannreceivedthePotterPrizeoftheAmericanMathematicalSocietyin1937;theBocherMemorialAwardin1938;theMeritoriousMedaloftheU.S.PresidentandtheU.S.NavyOutstandingCivilServiceAwardin1947;theFreedomMedalandtheU.S.President’sMedalofFreedomin1956FermiAward.

Anecdote

1.Once,atamathparty,ayoungmanenthusiasticallyapproachedhimandaskedhimaquestion.Helookedatitandreporteditcorrectly.Answer.Theyoungmanhappilyaskedhimtotellhimselftheeasyway,andcomplainedaboutthecumbersomenessofsolvinginfiniteserieswithothermathematicians.VonNeumannsaid:"Youmisunderstood,Ijustusedinfiniteseriestofindit."Itshowsthathehasextraordinarymentalarithmeticability.

2.Itissaidthatoneday,vonNeumannwasunsettledbyhiscolleaguesatthepokertable.Whileplayingcards,whilestillthinkingabouthissubject,he"lost"10yuaninembarrassment.Thiscolleague,whoisalsoamathematician,suddenlyhadaplantomakefunofhisfriend,soheusedthe5yuanhewontobuyacopyof"GameTheoryandEconomicBehavior"writtenbyvonNeumann,andputitTheremaining5yuanwaspostedonthecoverofthebooktoshowthathehad"beaten"the"gamblingeconomictheorist"andreallymadevonNeumann"noface".

3.DuringthedevelopmentoftheENIACcomputer.Severalmathematiciansgatheredtogethertodiscussmathematicalproblems,andtheycouldn'tthinkofasolutiontoacertainproblem.Someonedecidedtogohomewithadesktopcalculatortocontinuethecalculation.Inthemorningofthenextday,hewalkedintotheofficewithdarkeyecirclesandatiredface.Heproudlyshowedofftoeveryone:"Ihavebeencountingfromlastnightto4:30thismorning,andfinallyfound5specialsolutionstothatproblem.Theyare.Oneismoredifficultthanone!"Whilespeaking,vonNeumannopenedthedoorandcamein,"Whatquestionismoredifficult?"Althoughonlythelasthalfofthesentencewasheard,theword"harder"madehimimmediatelyexcited.Someonetoldhimaboutthetopic,theprofessorimmediatelythrewwhatheshoulddoinJava,andenthusiasticallysuggested:"Letuscountthese5specialsolutionstogether."

EveryoneIwanttoseetheprofessor's"sacredcalculation"skills.IsawvonNeumannstaringattheceiling,withoutsayingaword,quicklyenteringthestateof"enteringconcentration".Afterabout5minutes,Isaidthefirst4answers,andIwasthinkingaboutthe5th...Theyoungmathematiciancouldn'thelpitanymoreandcouldn'thelpbutblurtouttheanswer.VonNeumannwastakenaback,butdidnotanswerthequestion.Ittookanotherminutebeforehesaid:"Youareright!"

Themathematicianleftinareverentmood.Hethoughtteasingly:"Whatothercomputersareyoubuilding,ProfessorIsn’thismindjusta“super-high-speedcomputer”?”However,vonNeumannstayedinplace,fallingintohardthinking,unabletoextricatehimselfforalongtime.Someoneaskedhimsoftlyaboutthereason,andtheprofessorrepliedanxiously:"Iwaswonderingwhatmethodheusedtofigureouttheanswersoquickly."Hearingthis,everyonecouldn'thelplaughing:"Heusesdesktopcomputing.It'sbeenawholenight!"VonNeumannwastakenabackandthenlaughedheartily.

4.VonNeumann’sdrivingskillsareverypoor.Accidentsoftenoccur.Oncehecrashedthefrontofthecarandexplainedinthepolicestation:“I’mdrivingnormallyontheroad,andthetreeoutsidethewindowontherightIwaspassingbymycarataspeedof60milesperhour.Suddenly,atreestoodinfrontofmycar.Boom!"

5.JustbeforevonNeumannpassedawayGod,thetumorhasoccupiedhisbrain,butthememoryissometimesunbelievablygood.ThatdayUlamsatinfrontofhisbedandrecitedabookofThucydidesinGreek,theAdeniteheparticularlylikesattackingMelosHeremembersthestoryandPerile'sspeechveryfirmlyandwillcorrectUlam'smistakesandpronunciation.

SocialEvaluation

CognitiveAbility

HansBethe(NobelPrizeWinnerinPhysics):IhavesometimeswonderedwhetherabrainlikevonNeumann'sdoesnotindicateaspeciessuperiortothatofman.(IsometimeswonderwhetherabrainlikevonNeumannimpliestheexistenceofbiologicalspecieshigherthanhumans.)

DavidBlackwell:Hewasareallyremarkableman.HelistenedtometalkaboutthisratherobscuresubjectandintenminutesheknewmoreaboutitthanIdid.Hewasextremelyquick.AftertenminutesheknewmorethanIdid.Histhinkingwasreallyunusuallyquick.)

GeorgePólya:TheonlystudentofmineIwaseverintimidatedby.Hewassoquick.TherewasaseminarforadvancedstudentsinZürichthatIwasteachingandvonNeumannwasintheclass.Icametoacertaintheorem,andIsaiditisnotprovedanditmaybedifficult.VonNeumanndidn'tsayanythingbutafterfiveminutesheraisedhishand.WhenIcalledonhimhewenttotheblackboardandproceededtowritedowntheproof.AfterthatIwasafraidofvonNeumann.(HeistheonlyonewhomakesmefeelthatmyteacherstatusisthreatenedStudent,heissokeen.OnceIgavealecturetograduatestudentsinZurich,whenvonNeumannwasalsoattendingtheclass,ImentionedTherewasanunresolvedquestion.Fiveminuteslater,vonNeumannraisedhishand.WhenIcalledhim,hewentstraighttothepodiumandwrotedowntheproofoftheproblem.Sincethen,IhavespokentovonNeumann.Feelintimidated.)

EugeneWigner(NobelLaureateinPhysics):Hehasagreatsenseofhumor,becausehisabilitytotellstoriesandjokesisverypopularevenwithstrangers.HecanWhenitbecomessimpleandhappy,butneverfrivolousandstupid.VonNeumann'sextraordinarybrainneedstounderstandwhatmostpeoplelikeusdon'twanttounderstandorevendonotwanttounderstand.ThisfactaffectsvonNeumann’smoraljudgment.Onlyscientificerrorsanddisharmonywillmakehimangryorregretful.Whenanyonecommitsascientificerror,hewillcorrectothers’mistakeswithouthesitation.mistake.

HermanGoldstine:Oneofhisextraordinaryabilityisabsolutelyaccuratememory.AsfarasIknow,vonNeumannhastheabilitytoreadabookoranarticleHereciteedonewordwell,andwhat'smore,hecouldalsodoitmanyyearslaterwithoutanyhesitation.HecanalsotranslateitintoEnglishandreciteitinrealtimewithoutanyreductioninspeed.OnceIaskedhimhow"ATaleofTwoCities"startedtotesthisabilities.Afterawhile,hebegantorecitethefirstchapteruntilmaybe10or15minuteslaterIaskedhimtostop.

Achievementevaluation

MiklósRédei:Itseemsfairtosaythatiftheinfluenceofascientistisinterpretedbroadlyenoughtoincludeimpactonfieldsbeyondscienceproper,thenJohnvonNeumannwasprobablythemostinfluentialmathematicianwhoeverlived.(Ifyoutalkaboutwhoistossedbetweenscientificfieldsbecauseofmathematicswork,andeveryjobyoudocanapplymathematicsinitsfieldandhavealong-termimpact,vonNeumanncansayHeisthemostinfluentialmathematicianinhistory.)

Glimm:heisregardedasoneofthegiantsofmodernmathematics.(Heisgenerallyregardedasoneofthegiantsofmodernmathematics.Agiant.)

JeanDieudonné:thelastofthegreatmathematician.(Thelastoutstandingmathematician.)

PeterLax:ThemostPeoplewithscientificminds.