问题：﻿有没有「数学直觉」这种东西？ 都说数学是讲逻辑的，但我发现，其实很多数学题做起来是需要直觉的。比如几何题里面的辅助线，我完全不知道怎么画，高手来了，完全是凭直觉画一条线，然后就可以…
话题：﻿教育、数学、战争
回答：﻿题主如果对数学也有兴趣，可以参考T.Tao的这篇博客There’smoretomathematicsthanrigourandproofshttp://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/陶讲的是数学研究，可能和题主说的解题离得比较远，我稍微转化一下论述方式，看能不能回答题主的疑惑。数学的结构是讲求逻辑，严谨，严格的，但是解决数学问题（做题也好科研也好）需要的则远不止这些，其中就包括“直觉”，即对于数学问题直观的理解的想象。做好数学要两者并重，即培养直觉的同时学会严格论述的能力，同时通过严格论述的能力修正自己的直觉，强化自己的直觉。如果题主只是想做好题而不是学好数学，最方便的是多做，多记。中学数学没有新题，无非是各种题型的转化和组合，通过细致的归纳总结，完全可以熟能生巧，不需要特别的直觉。即使天赋上差一点，完全可以以勤补拙。注意是细致的归纳总结，不是一味的傻做。比如画辅助线，你可以找一本有答案的习题集，把需要画辅助线的题目跳出来，根据答案总结辅助线的类型，适用的情况（条件）。关于解题，TTao也有一篇博客Solvingmathematicalproblemshttp://terrytao.wordpress.com/career-advice/solving-mathematical-problems/墙里是不是看不到wordpress？一并转一下吧SolvingmathematicalproblemsProblemsolving,fromhomeworkproblemstounsolvedproblems,iscertainlyanimportantaspectofmathematics,thoughdefinitelynottheonlyone.Laterinyourresearchcareer,youwillfindthatproblemsaremainlysolvedbyknowledge(ofyourownfieldandofotherfields),experience,patienceandhardwork;butforthetypeofproblemsoneseesinschool,collegeorinmathematicscompetitionsoneneedsaslightlydifferentsetofproblemsolvingskills.Idohaveabookonhowtosolvemathematicalproblemsatthislevel;inparticular,thefirstchapterdiscussesgeneralproblem-solvingstrategies.Thereareofcourseseveralotherproblem-solvingbooks,suchasPolya’sclassic“Howtosolveit“,whichImyselflearntfromwhilecompetingattheMathematicsOlympiads.Solvinghomeworkproblemsisanessentialcomponentofreallylearningamathematicalsubject–itshowsthatyoucan“walkthewalk”andnotjust“talkthetalk”,andinparticularidentifiesanyspecificweaknessesyouhavewiththematerial.It’sworthpersistingintryingtounderstandhowtodotheseproblems,andnotjustfortheimmediategoalofgettingagoodgrade;ifyouhaveadifficultywiththehomeworkwhichisnotresolved,itislikelytocauseyoufurtherdifficultieslaterinthecourse,orinsubsequentcourses.Ifindthat“playing”withaproblem,evenafteryouhavesolvedit,isveryhelpfulforunderstandingtheunderlyingmechanismofthesolutionbetter.Forinstance,onecantryremovingsomehypotheses,ortryingtoproveastrongerconclusion.See“askyourselfdumbquestions“.It’salsobesttokeepinmindthatobtainingasolutionisonlytheshort-termgoalofsolvingamathematicalproblem.Thelong-termgoalistoincreaseyourunderstandingofasubject.Agoodruleofthumbisthatifyoucannotadequatelyexplainthesolutionofaproblemtoaclassmate,thenyouhaven’treallyunderstoodthesolutionyourself,andyoumayneedtothinkabouttheproblemmore(forinstance,bycoveringupthesolutionandtryingitagain).Forrelatedreasons,oneshouldvaluepartialprogressonaproblemasbeingasteppingstonetoacompletesolution(andalsoasanimportantwaytodeepenone’sunderstandingofthesubject).SeealsoEricSchechter’s“Commonerrorsinundergraduatemathematics“.Ialsohaveapostonproblemsolvingstrategiesinrealanalysis.There’smoretomathematicsthanrigourandproofsThehistoryofeverymajorgalacticcivilizationtendstopassthroughthreedistinctandrecognizablephases,thoseofSurvival,InquiryandSophistication,otherwiseknownastheHow,Why,andWherephases.Forinstance,thefirstphaseischaracterizedbythequestion‘Howcanweeat?’,thesecondbythequestion‘Whydoweeat?’andthethirdbythequestion,‘Whereshallwehavelunch?’(DouglasAdams,“TheHitchhiker’sGuidetotheGalaxy“)Onecanroughlydividemathematicaleducationintothreestages:The“pre-rigorous”stage,inwhichmathematicsistaughtinaninformal,intuitivemanner,basedonexamples,fuzzynotions,andhand-waving.(Forinstance,calculusisusuallyfirstintroducedintermsofslopes,areas,ratesofchange,andsoforth.)Theemphasisismoreoncomputationthanontheory.Thisstagegenerallylastsuntiltheearlyundergraduateyears.The“rigorous”stage,inwhichoneisnowtaughtthatinordertodomaths“properly”,oneneedstoworkandthinkinamuchmorepreciseandformalmanner(e.g.re-doingcalculusbyusingepsilonsanddeltasallovertheplace).Theemphasisisnowprimarilyontheory;andoneisexpectedtobeabletocomfortablymanipulateabstractmathematicalobjectswithoutfocusingtoomuchonwhatsuchobjectsactually“mean”.Thisstageusuallyoccupiesthelaterundergraduateandearlygraduateyears.The“post-rigorous”stage,inwhichonehasgrowncomfortablewithalltherigorousfoundationsofone’schosenfield,andisnowreadytorevisitandrefineone’spre-rigorousintuitiononthesubject,butthistimewiththeintuitionsolidlybuttressedbyrigoroustheory.(Forinstance,inthisstageonewouldbeabletoquicklyandaccuratelyperformcomputationsinvectorcalculusbyusinganalogieswithscalarcalculus,orinformalandsemi-rigoroususeofinfinitesimals,big-Onotation,andsoforth,andbeabletoconvertallsuchcalculationsintoarigorousargumentwheneverrequired.)Theemphasisisnowonapplications,intuition,andthe“bigpicture”.Thisstageusuallyoccupiesthelategraduateyearsandbeyond.Thetransitionfromthefirststagetothesecondiswellknowntoberathertraumatic,withthedreaded“proof-typequestions”beingthebaneofmanyamathsundergraduate.(Seealso“There’smoretomathsthangradesandexamsandmethods“.)Butthetransitionfromthesecondtothethirdisequallyimportant,andshouldnotbeforgotten.Itisofcoursevitallyimportantthatyouknowhowtothinkrigorously,asthisgivesyouthedisciplinetoavoidmanycommonerrorsandpurgemanymisconceptions.Unfortunately,thishastheunintendedconsequencethat“fuzzier”or“intuitive”thinking(suchasheuristicreasoning,judiciousextrapolationfromexamples,oranalogieswithothercontextssuchasphysics)getsdeprecatedas“non-rigorous”.Alltoooften,oneendsupdiscardingone’sinitialintuitionandisonlyabletoprocessmathematicsataformallevel,thusgettingstalledatthesecondstageofone’smathematicaleducation.(Amongotherthings,thiscanimpactone’sabilitytoreadmathematicalpapers;anoverlyliteralmindsetcanleadto“compilationerrors”whenoneencountersevenasingletypoorambiguityinsuchapaper.)Thepointofrigourisnottodestroyallintuition;instead,itshouldbeusedtodestroybadintuitionwhileclarifyingandelevatinggoodintuition.Itisonlywithacombinationofbothrigorousformalismandgoodintuitionthatonecantacklecomplexmathematicalproblems;oneneedstheformertocorrectlydealwiththefinedetails,andthelattertocorrectlydealwiththebigpicture.Withoutoneortheother,youwillspendalotoftimeblunderingaroundinthedark(whichcanbeinstructive,butishighlyinefficient).Soonceyouarefullycomfortablewithrigorousmathematicalthinking,youshouldrevisityourintuitionsonthesubjectanduseyournewthinkingskillstotestandrefinetheseintuitionsratherthandiscardthem.Onewaytodothisistoaskyourselfdumbquestions;anotheristorelearnyourfield.Theidealstatetoreachiswheneveryheuristicargumentnaturallysuggestsitsrigorouscounterpart,andviceversa.Thenyouwillbeabletotacklemathsproblemsbyusingbothhalvesofyourbrainatonce–i.e.thesamewayyoualreadytackleproblemsin“reallife”.Seealso:BillThurston’sarticle“Onproofandprogressinmathematics“;HenriPoincare’s“Intuitionandlogicinmathematics“;thisspeechbyStephenFryontheanalogousphenomenonthatthereismoretolanguagethangrammarandspelling;andKohlberg’sstagesofmoraldevelopment(whichindicate(amongotherthings)thatthereismoretomoralitythancustomsandsocialapproval).Addedlater:Itisperhapsworthnotingthatmathematiciansatallthreeoftheabovestagesofmathematicaldevelopmentcanstillmakeformalmistakesintheirmathematicalwriting.However,thenatureofthesemistakestendstoberatherdifferent,dependingonwhatstageoneisat:Mathematiciansatthepre-rigorousstageofdevelopmentoftenmakeformalerrorsbecausetheyareunabletounderstandhowtherigorousmathematicalformalismactuallyworks,andareinsteadapplyingformalrulesorheuristicsblindly.Itcanoftenbequitedifficultforsuchmathematicianstoappreciateandcorrecttheseerrorsevenwhenthoseerrorsareexplicitlypointedouttothem.Mathematiciansattherigorousstageofdevelopmentcanstillmakeformalerrorsbecausetheyhavenotyetperfectedtheirformalunderstanding,orareunabletoperformenough“sanitychecks”againstintuitionorotherrulesofthumbtocatch,say,asignerror,orafailuretocorrectlyverifyacrucialhypothesisinatool.However,sucherrorscanusuallybedetected(andoftenrepaired)oncetheyarepointedouttothem.Mathematiciansatthepost-rigorousstageofdevelopmentarenotinfallible,andarestillcapableofmakingformalerrorsintheirwriting.Butthisisoftenbecausetheynolongerneedtheformalisminordertoperformhigh-levelmathematicalreasoning,andareactuallyproceedinglargelythroughintuition,whichisthentranslated(possiblyincorrectly)intoformalmathematicallanguage.Thedistinctionbetweenthethreetypesoferrorscanleadtothephenomenon(whichcanoftenbequitepuzzlingtoreadersatearlierstagesofmathematicaldevelopment)ofamathematicalargumentbyapost-rigorousmathematicianwhichlocallycontainsanumberoftyposandotherformalerrors,butisgloballyquitesound,withthelocalerrorspropagatingforawhilebeforebeingcancelledoutbyotherlocalerrors.(Incontrast,whenuncheckedbyasolidintuition,onceanerrorisintroducedinanargumentbyapre-rigorousorrigorousmathematician,itispossiblefortheerrortopropagateoutofcontroluntiloneisleftwithcompletenonsenseattheendoftheargument.)Seethispostforsomefurtherdiscussionofsucherrors,andhowtoreadpaperstocompensateforthem.