资源描述：
《几何基础书评（庞加莱）》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、BOOKREVIEWSBULLETIN(NewSeries)OFTHEAMERICANMATHEMATICALSOCIETYVolume37,Number1,Pages77{78S0273-0979(99)00806-XArticleelectronicallypublishedonDecember21,1999Poincare'sreviewofHilbert'sFoundationsofgeometry,byDavidHilbertWhatarethefundamentalprinciplesofgeometry?
2、whatisitsorigin?itsnature?itsscope?Thesearequestionswhichhaveatalltimesengagedtheattentionofmathematiciansandthinkers,butwhichaboutacenturyagotookonanentirelynewaspect,thankstotheideasofLobachevskyandofBolyai.Foralongtimeweattemptedtodemonstratethepropositionknow
3、nasthepos-tulateofEuclid;weconstantlyfailed;weknownowthereasonforthesefailures.Lobachevskysucceededinbuildingalogicaledi ceascoherentasthegeometryofEuclid,butinwhichthefamouspostulateisassumedfalse,andinwhichthesumoftheanglesofatriangleisalwayslessthantworightang
4、les.Riemanndevisedanotherlogicalsystem,equallyfreefromcontradiction,inwhichthissumisontheotherhandalwaysgreaterthantworightangles.Thesetwogeometries,thatofLobachevskyandthatofRiemann,arewhatarecalledthenon-euclideangeome-tries.ThepostulateofEuclidthencannotbedemo
5、nstrated;andthisimpossibilityisasabsolutelycertainasanymathematicaltruthwhatsoever
6、afactwhichdoesnotpreventtheAcademiedesSciencesfromreceivingeveryyearseveralnewproofs,towhichitnaturallyrefusesthehospitalityoftheComptesrendus.Muchhasalreadybeenwrittenonthenon-eu
7、clideangeometries;oncetheyscan-dalizedus;nowwehavebecomeaccustomedtotheirparadoxes;somepeoplehavegonesofarastodoubtthetruthofthepostulateandtoaskwhetherrealspaceisplane,asEuclidassumed,orwhetheritmaynotpresentaslightcurvature.Theyevensupposedthatexperimentcouldgi
8、vethemananswer[250]tothisquestion.Needlesstoaddthatthiswasatotalmisconceptionofthenatureofgeometry,whichisnotanexperimentalscience.Butwhy,amongalltheaxiomsofgeometry,shouldthispostulatebetheonlyonewhichcouldbedeniedwithouto encetologic?Whenceshoulditderivethispri
9、vilege?Thereseemstobenogoodreasonforthis,andmanyotherconceptionsarepossible.However,manycontemporarygeometersdonotappeartothinkso.Inrecogniz-ingtheclaimsofthet
