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1、BULLETIN(NewSeries)OFTHEAMERICANMATHEMATICALSOCIETYS0273-0979(07)01168-8ArticleelectronicallypublishedonMay2,2007WHATISGOODMATHEMATICS?TERENCETAOAbstract.Somepersonalthoughtsandopinionsonwhat“goodqualitymath-ematics”isandwhetheroneshouldtrytodefinethist
2、ermrigorously.Asacasestudy,thestoryofSzemer´edi’stheoremispresented.1.ThemanyaspectsofmathematicalqualityWeallagreethatmathematiciansshouldstrivetoproducegoodmathematics.Buthowdoesonedefine“goodmathematics”,andshouldoneevendaretotryatall?Letusfirstconsid
3、ertheformerquestion.Almostimmediatelyonerealisesthattherearemanydifferenttypesofmathematicswhichcouldbedesignated“good”.Forinstance,“goodmathematics”couldrefer(innoparticularorder)to(i)Goodmathematicalproblemsolving(e.g.amajorbreakthroughonanimportantma
4、thematicalproblem);(ii)Goodmathematicaltechnique(e.g.amasterfuluseofexistingmethodsorthedevelopmentofnewtools);(iii)Goodmathematicaltheory(e.g.aconceptualframeworkorchoiceofno-tationwhichsystematicallyunifiesandgeneralisesanexistingbodyofresults);(iv)Go
5、odmathematicalinsight(e.g.amajorconceptualsimplificationortherealisationofaunifyingprinciple,heuristic,analogy,ortheme);(v)Goodmathematicaldiscovery(e.g.therevelationofanunexpectedandintriguingnewmathematicalphenomenon,connection,orcounterexam-ple);(vi)
6、Goodmathematicalapplication(e.g.toimportantproblemsinphysics,engineering,computerscience,statistics,etc.,orfromonefieldofmathe-maticstoanother);(vii)Goodmathematicalexposition(e.g.adetailedandinformativesurveyonatimelymathematicaltopicoraclearandwell-mo
7、tivatedargument);(viii)Goodmathematicalpedagogy(e.g.alectureorwritingstylewhichenablesotherstolearnanddomathematicsmoreeffectively,orcontributionstomathematicaleducation);(ix)Goodmathematicalvision(e.g.along-rangeandfruitfulprogramorsetofconjectures);(x
8、)Goodmathematicaltaste(e.g.aresearchgoalwhichisinherentlyinter-estingandimpactsimportanttopics,themes,orquestions);ReceivedbytheeditorsFebruary7,2007.2000MathematicsSubjectClassification.Primary00A30.c2007AmericanMathematicalSocietyReve
