资源描述：
《What is good mathematics-Terence Tao.pdf》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。
1、WHATISGOODMATHEMATICS?TERENCETAOAbstract.Somepersonalthoughtsandopinionsonwhat“goodqualitymathemat-ics”is,andwhetheroneshouldtrytodefinethistermrigorously.Asacasestudy,thestoryofSzemer´edi’stheoremispresented.1.ThemanyaspectsofmathematicalqualityWeallagreethatmathematiciansshou
2、ldstrivetoproducegoodmathematics.Buthowdoesonedefine“goodmathematics”,andshouldoneevendaretotryatall?Letusfirstconsidertheformerquestion.Almostimmediatelyonerealisesthattherearemanydifferenttypesofmathematicswhichcouldbedesignated“good”.Forinstance,“goodmathematics”couldrefer(inn
3、oparticularorder)to(i)Goodmathematicalproblem-solving(e.g.amajorbreakthroughonanimpor-tantmathematicalproblem);(ii)Goodmathematicaltechnique(e.g.amasterfuluseofexistingmethods,orthedevelopmentofnewtools);(iii)Goodmathematicaltheory(e.g.aconceptualframeworkorchoiceofnotationwhi
4、chsystematicallyunifiesandgeneralisesanexistingbodyofresults);(iv)Goodmathematicalinsight(e.g.amajorconceptualsimplification,ortherealisationofaunifyingprinciple,heuristic,analogy,ortheme);(v)Goodmathematicaldiscovery(e.g.therevelationofanunexpectedandin-triguingnewmathematicalp
5、henomenon,connection,orcounterexample);(vi)Goodmathematicalapplication(e.g.toimportantproblemsinphysics,engi-neering,computerscience,statistics,etc.,orfromonefieldofmathematicstoanother);(vii)Goodmathematicalexposition(e.g.adetailedandinformativesurveyonaarXiv:math.HO/0702396v1
6、13Feb2007timelymathematicaltopic,oraclearandwell-motivatedargument);(viii)Goodmathematicalpedagogy(e.g.alectureorwritingstylewhichenablesotherstolearnanddomathematicsmoreeffectively,orcontributionstomath-ematicaleducation);(ix)Goodmathematicalvision(e.g.along-rangeandfruitfulpr
7、ogramorsetofconjectures);(x)Goodmathematicaltaste(e.g.aresearchgoalwhichisinherentlyinterestingandimpactsimportanttopics,themes,orquestions);(xi)Goodmathematicalpublicrelations(e.g.aneffectiveshowcasingofamathe-maticalachievementtonon-mathematicians,orfromonefieldofmathematicsto
8、another);(xii)Goodmeta-mathematics(e.g.advancesinthefoundatio
