Anon-heuristicmethodisonethatmaybeofgreatuseinshoringupoursensethatas - PDF document

Anon-heuristicmethodisonethatmaybeofgreatuseinshoringupoursensethatastatementisplausibleoncewehavethestatementinmind,butisnotparticularlygoodatdiscoveringsuchstatementsforus.Inwhatfollows,I'llbediscussingeachofthesemodesinturn,notingthatthisthree-partdistinctionissometimesblurredbythefactthatallthreecanworksurprisinglywelltogether.Amongotherthings,I'llbespecicallythinkingaboutwhatmighthavebeenthemotivationforLeonhardEulertohavecomeupwithacertaincuriousconjecture. LeonhardEuler(1707-1783)1ReasoningfromConsequenceThisiscapturedbythemaxim:If(A)impliestruethingswegaincondencein(A).Dependingupontheparticularwayitiscast,itissometimesreferredtoasInduction,orExperimentalconrmation,or\Inferentialfallacy."Hereistheexamplewe'llfocuson.3 PolyadiscussesthefollowingconjectureofEuler1:Anynumberoftheform3+8n(fornpositive)isthesumofasquareandthedoubleofaprime2:(A)3+8n=a2+2p:Thisisstilltodayjustaconjecture,neitherprovennordisproven.Howwouldyouhaverstdiscoveredsuchastatement,asbeingpotentiallytrue?Havingcomingupwiththestatement,howwouldyougarnerevidenceforitstruth?Howwouldyouaugmentordiminishitslevelplausibility?Thatis,withoutactuallyprovingit.Ofcourse,facedwithsuchaproblem,therstthingonemight|perhapsshould|doistotestitnumerically:11=12+2
519=32+2
527=12+2
1335=12+2
17=32+2
13=52+2
5:::Now,Eulerbecameinterestedinthisconjecture|Polyaexplains|becausebyassumingit,Eulercouldprove:Anynumberisasumofthreetrigonalnumbers:(B)n=x(x+1) 2+y(y+1) 2+z(z+1) 2;aresulthebelievedtobetrueandhadbeenpreviouslyinterestedin3.ItisintriguingtofollowEuler's(strange,Ithink)trainofthought:whatgothimtothinkthatthe(A)hewasspecicallyinterestedinwouldbemademoreplausiblebyvirtueofitsimplyingthe(B)above?4Thislatterstatement,(B),isaspecialcaseofPierredeFermat'spolygonalnumber\theorem,"andeventhoughEulerbelieved(B)tobetrue,no(published)proofofitexistedatthetime;(B)was 1ThisisdiscussedasanexampleofVericationofaconsequence:page3ofVol.IIofMathematicsandPlausibleReasoning:Patternsofplausibleinference.Fortheoriginaltext,seetheAfterwordbelow.2Eulercouldincluden=0inhisassertion,sinceheallowed1tobeaprime|therebysidingwithmyfather,who,wheneverhewantedtogetmygoat,wouldplayfullyaskmetodefendmybizarrecontentionthattherstprimeis2.SinceIhaveremoved1asprime,fromtheinitialconjectureconceivedbyEulerIhaveever-so-slightlystrengthenedit.3Numbersoftheformn(n+1)=2arecalledtrigonalsincetheycanbethoughtofascountinganarrayofpointsintheplanethathaveintegralcoordinatesandform|i.e.,haveastheirconvexclosure|anisoscelesright-angletriangle.4AsketchofwhyAimplies(B)isgivenintheafterwordbelow.4 Inviewofthisdistinctionbetweenthetwomodesofthought,itishardlystrangethatmodicationsofformulationswhichmaybeequivalenttoeachotherinthecalculusoflogic,maybecomequitealientoeachotherinthecalculusofplausibility.Afamousexampleofsuchamodicationispassagetothecontrapositive,asinthephilosophicalconundrum,duelargelytoCarlGustavHempel.ThisissometimescalledHempel'sparadox,ortheravenparadox,andstartswiththeobservationthat\logicallyspeaking"thetwostatements:AllravensareblackandNonon-blackobjectisaravenaregenuinelyequivalent;andyetthenatural(reasoningfromconsequences)wayofcollectingem-piricalevidencefortherststatementistondravenafterravenandcheckwhetherornottheyareblack,whereasthecorrespondingstrategyforthesecondformulationistolookforobjectsthataren'tblack,andthencheckthattheyaren'travens.Nowtherststrategytogaincondenceintheassertionseemsquixoticperhaps,butvaguelysanetous,whilethesecondisutterlyludicrous.(Theessenceofthis\paradox,"asfarasIunderstandit,isthateventhoughthetwostatementsaboveareequivalentfromthevantage-pointofthecalculusoflogic,theypresentquitedierentnaturalstrategiesfor\reasoningfromconsequence.")AtypicalmathematicalanaloguetoHempel'sconundrum|whichshedssomelightontheinitialissue|mightbethecomparisonofthefollowingtwomodesofbuildingcondenceintheRiemannhypothesis,which|formulatedinthetraditionalway|asserts(stillconjecturally,ofcourse)thatallthezeroesoftheRiemannzeta-function,(s),eitherlieontherealline(inwhichcasetheyoccuratnegativeevenintegervaluesofs:thesearecalledthetrivialzeroesof(s)sincethequestionofvanishing|ornonvanishing|ofthezetafunctiononthereallineisknown)orelsethezeroeslieonthelineRe(s)=1=2inthecomplexplane.TheHempelconundrum,giventhisformulation,wouldbetonotethefactthatthenaturalreasoningfromconsequencemodeofcollectingevidenceforthehypothesisasstated,andofitscontrapositive,wouldbe|respectively:Finda(nontrivial)zerooftheRiemann-functionandcheckthatitactuallyliesonthelineRe(s)=1 2,or:ndapoints0inthecomplexplanethatis(notatrivialzeroand)isothelineRe(s)=1 2,andcheckthat(s0)6=0.Thereareafewthingstodiscuss,evenwithregardtothissomewhatfrivolousexample.Asforthesecondstrategyabove,ifweweretochooseans0outsidethecriticalstrip,wewouldlearnabsolutelynothingwehadn'tknown|sincewealreadyknowthatallthetrivialzeroeslieinthecriticalstrip.Evenkeepingtothecriticalstrip,however,wedohavethepriorknowledgethatthesetofzeroesisdiscrete,sothechancesofhittingonazerobyarandomchoiceofapointis:::well...zero.Incontrast,thedelicacyoftherststrategy|requiringsomethingtolieonagivenline|isaverydemandingtest.Inbrief,thisintricatenetworkof\priorassessments"controlthecalculusofplausibilityrelatedtothesestrategies.Whatmakesreasoningfromconsequencesomulti-strandedisillustratedbythisexampleoftheRiemannHypothesis,wheretherearemanyequivalentformulationsofthisverysamehypothesis,andeachformulationprovidesuswithyet6 theother,andtheyeachhavetheirownidiosyncraticrelationshipto\time."LetusreturnoncemoretoPolya'sexample{i.e.,Euler'sConjecturethatanynumberoftheform3+8nisasquareplustwiceaprime.Eventhough,aswe'vejustmentioned,Eulerachievedasenseofitsplausibilityfromwhatwe'recallingreasoningfromconsequence,thereisanotherplausibilityrouteforthissameconjecture.Onemightcometobelievethissameconjecture|oratleastsomethingqualitativelylikethatconjecture|viaaheuristicmethodthatproposesthatonceonetakesaccountofalltheknownconstraints,thedataisrandom.2ReasoningfromRandomnessAsmentionedabove,thismodeofreasoningcanbecapturedby,thefollowingsentiment:Weknowalltherelevantsystematicconstraintsinthephenomenathatwearecurrentlystudying,and:::therestisrandom5.Itisasomewhathubristictypeofreasoning,butitis|asI'msureyouagree|acommon,andperfectlynatural,wayofthinking;itisveryoftenapowerfulmethodthatleadstoformulatinghypothesesthat|ifnotalwaystrue|atleastoftenrepresent\currentbestguesses."Thismethod,incontrastto\reasoningbyconsequence,"isgenuinelyheuristic:whenitisapplicable,itdoesindeedpresentuswithfairlypreciseformulations.Hereisasimpleexampleofaproblemwhichwillillustratethepower,andtheessentiallimitationsofthishubristicmethod.Leta;b;cbeatripleofpositiveintegers.ConsiderthediophantineequationA+B=CwhereAisallowedtobeanypositiveintegerthatisaperfecta-thpower,Baperfectb-thpowerandCaperfectc-thpower.(So,forexample,ifa=b=c=2weareconsideringPythagoreantriples.)LetXbealargepositiveinteger,andN(X)bethenumberofsolutionsofourdiophantineequationwithCX.WhatcanwesayaboutthebehaviorofN(X)asafunctionoftheboundX?Andhereisaroughargumentthatmightleadyoutosomekindofconjectureregardingthisproblem.Itisinagenreofspeculationthatalmostallmathematiciansmusthaveengagedin,atonetimeoranother,inthevocabularyoftheireldsofinterest.(ForsimplicityIwillignorepositiveconstantsindependentofXthatariseinourerrorestimateseitherasmultiplicativefactorsorsimplyasconstants.)Herearepossiblestepsinourdeliberation: 5Weformthe[measure]spacecomprisingthepossibleoutcomesweareinterestedin,subjecttoallconstraintsthatwehappentoknowof,andthenweputwhatweconsidertobesomekindof'even-handed'probabilitymeasureonthisspace.Thisissometimestaggedasanapplicationoftheprincipleofinsucientreason.8 shotsatthis.Sotheexpectednumberofsuccesseswillbe1 XtimesX1 a+1 b+1 c,or:X1 a+1 b+1 c�1:Toblurthingsabit,giventhatwehavebeenarguingquitenaively,wemightconjecture:When1 a+1 b+1 c�1;weshouldget:X1 a+1 b+1 c�1�No(X)X1 a+1 b+1 c�1+forany�0,andforX��0(withtheimpliedconstantin\��"dependingon).When1 a+1 b+1 c1;theaboveestimatewouldgiveusadecreasingnumberofhitsasXtendstoinnity;whichdoesn'tmakemuchsenseatall,butweinterpretitassuggestingConjecture1Fixingexponentsa;b;csatisfyingtheinequality1 a+1 b+1 c1;thereareonlynitelymanysolutionstothediophantineproblemUa+Vb=Wcwith(U;V;W)relativelyprimepositivenumbers.WeseetheclassicalLastTheoremofFermatasgivingagooddealmorepreciseinformationthantheaboveconjectureforthecasesa=b=c&#x]TJ/;༕ ;.9;‘ ;&#xTf 1;
.51; 0 ;&#xTd [;3.Thisillustratesthestructuralshortcomingofthisprobabilisticheuristic:itisquintessentiallyprobabilistic,andcouldnotgetonetomakeaspreciseaconjectureasFermat'sLastTheorem,eventhoughitmightoer,asplausibleguess,somearmationofthequalitativeaspectofthatTheorem7.AsImentioned,everymathematicianmusthavesomefavoriteapplicationsofreasoningfromran-domness.Innumbertheory,mycurrentfavoriteistheCohen-Lenstraheuristicthatgiveusguessesfortheaveragevaluesofidealclassgroupsovervariousrangesofnumberelds.Thisisobtainedbyimaginingthethought-experimentoffabricatinganidealclassgroupbyarandomprocessintermsofitsgeneratorsandrelations,subjecttothepriorconstraintsthatre ecteverythingweknowaboutthemannerinwhichtheidealclassgroupappears.Sofar,theCohen-Lenstraheuristicsseemtocheckoutwithnumericalcomputations,andtheseheuristicsareregardedassucientlyplausible,thattheyhavearmplaceinthetoolkitofconjecturesthatarehelpful|eventhoughnotyetproven|guidesforre ectionsandexperimentsinthatbranchofnumbertheory8. 7I'veleftoutoftheabovediscussionthecaseofequality1 a+1 b+1 c=1,whichinvolvesjustahandfulofpossibilities:(3;3;3);(2;3;6);(2;4;4)andtheirpermutations;eachofthemhaveinterestingstories.ThediscussionaboveinmoregeneralformatmotivatestheworkofMasserandOesterlewiththeirwonderful,sweeping,ABCconjecture;see[?]).8Morerecently,therehavebeenheuristics,somewhatinthespiritofCohen-Lenstra,thatpredictthedensitiesofgivenranksofp-Selmergroupsovertheclassofallellipticcurvesoveragivennumbereld;theseheuristicsareduetoBjornPoonenandEricRains.10 mathematicalthoughtissaturatedwith,andverymuchcoloredby,analogiesofallsorts.Here,then,aretwosomewhatdierentbrandsofanalogies:Analogybyexpansioniswhereonehasaconcept,oraconstellationofconcepts,oratheory,andonewantstoexpandthereachoftheseconcepts,retainingtheirstructureassomesortoftemplate.Thisisoftenreferredtosimplyas\generalization"butthatterm|generalization|is,Ithink,moreusefulifitwereallowedtobealooserdescriptive,including,aswell,someoftheothertypesofanalogicaloperationsthatI'llbelisting.Analogybyexpansionmayhavetheappearance,afterthefact,ofbeingaperfectlynatural\analyticcontinuation,"sotospeak,ofaconcept|suchasthedevelopmentofzeroandnegativenumbersasanexpansionofwholenumbers,andfromthere:rationalnumbers,etc.BUT,theactitselfmaybe,atthetimeandevensomewhatafterwards,havetheshockvalueofafundamentalchange.EventhoughGrothendiecktopologiesarehalfacenturyold,andare(butonlyafteryou'velearnedthetheory!)anutterlydirectexpansionofthemoreclassicalnotionoftopology|anexpansionthatgetstotheessenceofthatclassicalconcept|thereisstillthethrillofitasprovidingaradicalreguringofwhatitmeanstobeatopology.Thereis,however,amoremodestversionofanalogybyexpansionthatweall,Ibelieve,constantlydo:startingfroma(B)thatweeitherknowtobetrue,oratleastrmlybelievetobetrue,welookaroundforimprovements(A)thatarejustabitmoregeneralthan(B),forwhichwecanshowthat(A)=)(B):Evenrelativelysmallimprovements,(A)'sthatarenomorethan(B)+,sotospeak,arefairgame.Subjecttonopalpablecounter-indication,theprovedimplicationconfersamodicumofbeliefin(B)+.ThisisinthestyleofAmazon'sIfyoulikeXyou'lllikeY,butheretakestheform:Ifyoubelieve(B)whydon'tyoubelieve(B)+?AsIhopewillbeclearfromtheAfterword(Section??)below,itisthisbelief-expansionattitudethat(Ithink)isalsoaplausibleaccountofhowEulercametohisConjecture.Analogyas'Rosettastone'iswhereonesubject,orbranchofasubject|withitsspecicvocabulary|isperceivedasrepresentingsomethingofamodelusefultopredictwhatmightbehappeninginanothersubject,orbranch,thislattersubjectoftenhavingquitedierentvocabulary,axioms,andgeneralset-up.HereisAndreWeil'sfamousparagraphonanalogy|expressingasentimentthatIrmlydisagreewith;neverthelessIkeepquotingandre-quotingit:Nothingismorefruitful|allmathematiciansknowit|thanthoseobscureanalogies,thosedisturbingre ectionsofonetheoryonanother;thosefurtivecaresses,thosein-explicablediscords;nothingalsogivesmorepleasuretotheresearcher.Thedaycomeswhenthisillusiondissolves:thepresentimentturnsintocertainty;theyokedtheoriesrevealtheircommonsourcebeforedisappearing.AstheGitateaches,oneachievesknowledgeandindierenceatthesametime.WhatIbelievehasbeenshowntobetrue,timeaftertime,inthedevelopmentofmathematicsisthat\yokedtheoriesrevealtheircommonsource."Thatis,analogiesbetweentwodierenttheories12 insupportofageneralassertionaretellingconsequences.Innumbertheory,forexample,therearegeneralconjecturesforwhichanimmenseamountofnumericaldatahasbeencollectedthatactuallydonot(atleastsignicantly)supportthegeneralassertion,andinfactwouldnaivelysuggestadierentqualitativeguess:::andyetwestill(atleastcurrently)believethegeneralconjectures.Foranexampleofthissee[?].Reasoningbyrandomnesshasthedangerthatwemaynotbetakingintoconsiderationallsystem-aticbehaviorrelevanttothephenomenawearestudying.Neverthelessithastwogreatadvantages:itisastartingposition,abestcurrentguess,worthcontemplatingtogetwhatmightbethelayoftheland,andevenifnotaccurate,itisoftenananalysisthatseparatesthesupposedrandomaspects(whichmayshowupas`errorterms")fromthemoreregularaspects(whichmayshowupas\dominantterms")ofthephenomena.Innumbertheory,thoseerrortermsmaythenalsohaveprofoundstructure(e.g.,See[?])).Butreasoningbyanalogyis,Ithink,thekeystone:itispresentinmuch(perhapsall)dailymathe-maticalthought,andisalsooftentheinspirationbehindsomeofthemajorlong-rangeprojectsinmathematics.And,AndreWeilwasrightwhenhesaid:\nothingalsogivesmorepleasuretotheresearcher."InnumbertheorytheLanglandsProgramisoneofthegrandanalogies|currentlybeingvigorouslypursued|connectingrepresentationtheory,algebraicgeometry,andarithmetic.Theanalogybetweennumbereldsandfunction-eldsofonevariableoverniteeldsisamoreelementary,andolderexample.Thisanalogyviewsthetwotypeofeldswementionedasasingleentity(calledaglobaleld)thatistreatableinauniedway9.5Variantsof`plausible'InourdiscussionofEuler'sConjecture,IhopethatI'vemadeaconvincingargumentthatonecould(andEulermighthave)cometobelievehisconjecturefromamixofthethreebrandsofplausiblereasoningthatIhadlabeledasseparatecategoriesatthebeginningofmyessay(consequence,randomness,andanalogy).Myfeelingisthatifwethinkthroughthehistoryofanyofourpersonalinvolvementwithanymathematicalissuethatisimportanttous,allresourcesavailabletouswillbeplayingsomeroleintheproceedings.Thewaytheseresourcesinteractmaybecomplex,andmaybekey. 9But,aswithallgreatanalogies,itsimperfectionssparkle,raisingquestionsthatmayleadtofuturetheories,fardeeperthantheoneswecurrentlyareathomewith:Whatisthefullstorythatconnectstheniteprimesofanumbereldtoitsarchimedeanprimes,its\primesatinnity"?Whatisthefullanalogueinthecontextofnumbereldsofthegenusofafunctioneld(ofonevariable)overaniteeld?Isthereananalogue,inthecontextofnumberelds,oftheproductofthesmoothprojectivemodelsoffunctionelds(ofonevariable)overaniteeld?14 6Afterword:Euler'sConjectureimpliesGauss's\EurekaTheo-rem."RecallthatwewanttoassumeEuler'sConjecturethatanyintegern0occursinanequation(A)withaanintegerandpaprimenumber,(A)3+8n=a2+2p;andprove(whatwilleventuallybeGauss'sTheorem;i.e.,)thatanyintegern0isexpressibleasasumofthreetrigonalnumbers.Theproofofthisisgiveninaseriesofstepsandhints.1.Foranyequationoftheform(A)above,thenumberaisodd,andpisoftheform4t+1.(Proof:worktheequation(A)modulo8.)2.Iftheprimepoccursinanequationoftheform(A)thenpisexpressibleasasumoftwosquares:p=u2+v2:(ThisisFermat'sTheorem.)3.Ifanynumbern0occursinanequationoftheform(A)thenanynumbern0occursinanequationoftheform(A0)3+8n=a2+b2+c2witha;b;coddintegers.(Proof:Ifp=u2+v2takeb:=u+v;andc:=u�v;thenworktheequation(A0)modulo8toshowoddnessofa;b;c.)Asusual,refertothepropositionthatassertstheabovefactforanyn0as\(A0)."4.Writea=2x+1;b=2y+1;c=2z+1;forx;y;zintegers;computetoget:(B)n=x(x+1) 2+y(y+1) 2+z(z+1) 2;andgoingtheotherway,showthat(B)=)(A0).5.Concludethat(A)=)(A0)()(B).WecannowspeculatemoreexactlyaboutEuler'splausibilitythinkinghere.Eulerbelieved(B),andsurelyknewthat(B)islogicallyequivalentto(A0).So,ofcourse,hebelieved(A0),andheknewthatthepassagefrom(A)to(A0)consistedinnothingmorethanreplacing,inthesecontexts,thesetofnumbersthatareoftheform2u2+2v2suchthatu2+v2isanoddprimenumberby16