几何基础书评(庞加莱)

几何基础书评(庞加莱)

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1、BOOKREVIEWSBULLETIN(NewSeries)OFTHEAMERICANMATHEMATICALSOCIETYVolume37,Number1,Pages77{78S0273-0979(99)00806-XArticleelectronicallypublishedonDecember21,1999Poincare'sreviewofHilbert'sFoundationsofgeometry,byDavidHilbertWhatarethefundamentalprinciplesofgeometry?whatisitsorigin?itsnature?itssc

2、ope?Thesearequestionswhichhaveatalltimesengagedtheattentionofmathematiciansandthinkers,butwhichaboutacenturyagotookonanentirelynewaspect,thankstotheideasofLobachevskyandofBolyai.Foralongtimeweattemptedtodemonstratethepropositionknownasthepos-tulateofEuclid;weconstantlyfailed;weknownowthereason

3、forthesefailures.Lobachevskysucceededinbuildingalogicaledi ceascoherentasthegeometryofEuclid,butinwhichthefamouspostulateisassumedfalse,andinwhichthesumoftheanglesofatriangleisalwayslessthantworightangles.Riemanndevisedanotherlogicalsystem,equallyfreefromcontradiction,inwhichthissumisontheothe

4、rhandalwaysgreaterthantworightangles.Thesetwogeometries,thatofLobachevskyandthatofRiemann,arewhatarecalledthenon-euclideangeome-tries.ThepostulateofEuclidthencannotbedemonstrated;andthisimpossibilityisasabsolutelycertainasanymathematicaltruthwhatsoever

5、afactwhichdoesnotpreventtheAcademiedesSc

6、iencesfromreceivingeveryyearseveralnewproofs,towhichitnaturallyrefusesthehospitalityoftheComptesrendus.Muchhasalreadybeenwrittenonthenon-euclideangeometries;oncetheyscan-dalizedus;nowwehavebecomeaccustomedtotheirparadoxes;somepeoplehavegonesofarastodoubtthetruthofthepostulateandtoaskwhetherrea

7、lspaceisplane,asEuclidassumed,orwhetheritmaynotpresentaslightcurvature.Theyevensupposedthatexperimentcouldgivethemananswer[250]tothisquestion.Needlesstoaddthatthiswasatotalmisconceptionofthenatureofgeometry,whichisnotanexperimentalscience.Butwhy,amongalltheaxiomsofgeometry,shouldthispostulateb

8、etheonlyonewhichcouldbedeniedwithouto encetologic?Whenceshoulditderivethisprivilege?Thereseemstobenogoodreasonforthis,andmanyotherconceptionsarepossible.However,manycontemporarygeometersdonotappeartothinkso.Inrecogniz-ingtheclaimsofthet

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