小朋友的涂鸦（四）：一个规划的大纲

CommeAppeléduNéant—AsIfSummonedfromtheVoid:TheLifeofAlexandreGrothendieckAllynJackson1196NOTICESOFTHEAMSVOLUME51,NUMBER10ADifferentWayofThinkingDansletravaildedécouverte,cetteat-tentionintense,cettesollicitudeardentesontuneforceessentielle,toutcommelachaleurdusoleilpourl’obscureges-tationdessemencesenfouiesdanslaterrenourricière,etpourleurhumbleetmiraculeuseéclosionàlalumièredujour.Intheworkofdiscovery,thisintenseat-tention,thisardentsolicitude,areanes-sentialforce,justlikethewarmthofthesunfortheobscuregestationofseedscoveredinnourishingsoil,andfortheirhumbleandmiraculousblossom-inginthelightofday.—RécoltesetSemailles,pageP49Grothendieckhadamathematicalstyleallhisown.AsMichaelArtinoftheMassachusettsInsti-tuteofTechnologycommented,inthelate1950sand1960s“theworldneededtogetusedtohim,tohispowerofabstraction.”NowadaysGrothen-dieck’spointofviewhasbeensothoroughlyab-sorbedintoalgebraicgeometrythatitisstandardfareforgraduatestudentsstartinginthefield,manyofwhomdonotrealizethatthingswereoncequitedifferent.NicholasKatzofPrincetonUni-versitysaidthatwhenasayoungmathematicianhefirstencounteredGrothendieck’swayofthink-ing,itseemedcompletelydifferentandnew.Butitishardtoarticulatewhatthedifferencewas.AsKatzputit,thechangeinpointofviewwassofun-damentalandprofoundand,onceadopted,socompletelynatural“thatit’ssortofhardtoimag-inethetimebeforeyouthoughtthatway.”AlthoughGrothendieckapproachedproblemsfromaverygeneralpointofview,hedidsonotforgenerality’ssakebutbecausehewasabletousegen-eralityinaveryfruitfulway.“It’sakindofapproachthatinlessgiftedhandsjustleadstowhatmostpeoplewouldsayaresterilegeneralities,”Katzcommented.“Hesomehowknewwhatgeneralthingstothinkabout.”Grothendieckalwayssoughtthepreciselevelofgeneralitythatwouldprovidepreciselytherightleveragetogaininsightintoaproblem.“Heseemedtohavetheknack,timeaftertime,ofstrippingawayjustenoughsothatitwasn’taspecialcase,butitwasn’tavacuumeither,”com-mentedJohnTateoftheUniversityofTexasatAustin.“It’sstreamlined;thereisnobaggage.It’sjustright.”OnestrikingcharacteristicofGrothendieck’smodeofthinkingisthatitseemedtorelysolittleonexamples.Thiscanbeseeninthelegendoftheso-called“Grothendieckprime”.Inamathematicalconversation,someonesuggestedtoGrothendieckthattheyshouldconsideraparticularprimenum-ber.“Youmeananactualnumber?”Grothendieckasked.Theotherpersonreplied,yes,anactualprimenumber.Grothendiecksuggested,“Allright,take57.”AllynJacks[e-mail]o-partarticleaboutthelifeofAlexandreGrothendieck.ThefirstpartofthearticleappearedintheOctober2004issueoftheNotices.NOVEMBER2004NOTICESOFTHEAMS1197ButGrothendieckmusthaveknownthat57isnotprime,right?Absolutelynot,saidDavidMumfordofBrownUniversity.“Hedoesn’tthinkconcretely.”ConsiderbycontrasttheIndianmathematicianRamanujan,whowasintimatelyfamiliarwithprop-ertiesofmanynumbers,someofthemhuge.ThatwayofthinkingrepresentsaworldantipodaltothatofGrothendieck.“Hereallyneverworkedonexam-ples,”Mumfordobserved.“Ionlyunderstandthingsthroughexamplesandthengraduallymakethemmoreabstract.Idon’tthinkithelpedGrothendieckintheleasttolookatanexample.Hereallygotcon-trolofthesituationbythinkingofitinabsolutelythemostabstractpossibleway.It’sjustverystrange.That’sthewayhismindworked.”NorbertA’CampooftheUniversityofBaselonceaskedGrothendieckaboutsomethingrelatedtothePlatonicsolids.Grothendieckadvisedcaution.ThePlatonicsolidsaresobeautifulandsoexceptional,hesaid,thatonecannotassumesuchexceptionalbeautywillholdinmoregeneralsituations.OnethingGrothendiecksaidwasthatoneshouldnevertrytoproveanythingthatisnotalmostob-vious.Thisdoesnotmeanthatoneshouldnotbeambitiousinchoosingthingstoworkon.Rather,“ifyoudon’tseethatwhatyouareworkingonisalmostobvious,thenyouarenotreadytoworkonthatyet,”explainedArthurOgusoftheUniversityofCaliforniaatBerkeley.“Preparetheway.Andthatwashisapproachtomathematics,thateverythingshouldbesonaturalthatitjustseemscompletelystraightforward.”Manymathematicianswillchooseawell-formulatedproblemandknockawayatit,anapproachthatGrothendieckdisliked.Inawell-knownpassageofRécoltesetSemailles,hedescribesthisapproachasbeingcomparabletocrackinganutwithahammerandchisel.Whathepreferstodoistosoftentheshellslowlyinwater,ortoleaveitinthesunandtherain,andwaitfortherightmomentwhenthenutopensnaturally(pages552–553).“SoalotofwhatGrothendieckdidlookslikethenaturallandscapeofthings,becauseitlookslikeitgrew,asifonitsown,”Ogusnoted.Grothendieckhadaflairforchoosingstriking,evocativenamesfornewconcepts;indeed,hesawtheactofnamingmathematicalobjectsasanin-tegralpartoftheirdiscovery,asawaytograspthemevenbeforetheyhavebeenentirelyunderstood(R&S,pageP24).Onesuchtermisétale,whichinFrenchisusedtodescribetheseaatslacktide,thatis,whenthetideisneithergoinginnorout.Atslacktide,thesurfaceofthesealookslikeasheet,whichevokesthenotionofacoveringspace.AsGrothen-dieckexplainedinRécoltesetSemailles,hechosethewordtopos,whichmeans“place”inGreek,tosuggesttheideaof“the‘objectparexcellence’towhichtopologicalintuitionapplies”(pag