数学模型与现实Word文件下载.docx
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数学模型与现实Word文件下载.docx
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Abstract
Thispaperexaminesthenatureandfunctionofmathematicalmodels,usingillustrationsfromcosmology,spacegeometryandatomicphysics.Mathematicalmodelsenableustomakeprecisecalculationsandpredictions;
theyserveasanalogiesandconceptualframeworksthatleadtonewdiscoveries;
andtheybridgethegapbetweenappearanceandreality.Theirsuccessimpliesthattheuniversehasamathematicalstructure.However,onemustbecarefulnottoconfusemodelsofrealitywithrealityitself.Avarietyofmodelscanrepresentthesamedata;
anymodelcanbegivendifferentphysicalinterpretations.Thechoiceofamodelanditsinterpretationdependslargelyonone'
sworldview
1.Introduction
Mathematicalmodelsplayalargeroleinscience.Inthispaperweshallinvestigateanumberofwaysinwhichmodelshavebeenused.Weshallillustratethesebyexamininganumberofhistoricalexamples,takenfromcosmology,spacegeometry,andatomictheory.Weshallexaminethestrengthsandweaknessofmathematicalmodels.
2.Whatisa“Model”?
Whatdowemeanbyamathematical“model”?
Webster’sNinthNewCollegiateDictionary(1989)givesadozenorsodefinitionsof“model”.Amongthese,themostpertinentforourpurposesarethefollowing:
4.aminiaturerepresentationofsomething.
11.adescriptionoranalogyusedtohelpvisualizesomething(asanatom)thatcannotbedirectlyobserved.
12.asystemofpostulates,data,andinferencespresentedasamathematicaldescriptionofanentityorstateofaffairs.
TheOxfordAmericanDictionaryandLanguageGuide(1999)givesafewfurtherdefinitions,including:
2.asimplified(oftenmathematical)descriptionofasystem,etc.,toassistcalculationsandpredictions.
Fromthesedefinitionsweshalltentativelytakeamodeltobeasimplified,oftenmathematical,representationoranalogyofsomeaspectofreality,forthepurposesofdescriptionand/orcalculation.
Modelsinsciencecomeinawidevarietyofforms.IanBarbour(1974:
29),inhisextensivediscussionofmodels,listsfourtypesofmodels:
1.Materialmodels.Thesearesimplified,scaled-down(orup)miniatures(e.g.,windtunnels,hydrodynamicmodels)oranaloguemodels(e.g.,anelectriccircuithasthesamebehaviorasamechanicalsystemofsprings&
dampers).Theyareusefulwhenitistoodifficulttoexperimentontheactualsystemorwhenthemathematicalequationsareunknownortoohardtosolve.
2.Mathematicalmodels.Thesearesetsofequations,usingsymbolicrepresentationsofquantitativevariablesinsimplifiedphysicalorsocialsystems(e.g.,populationgrowth,economics).Theirchieffunctionisprediction.
3.Logicalmodels.Alogicalmodelisaparticularsetofentitiessatisfyingagivensetofaxioms.Forexample,pointsandlinesingeometryisalogicalmodelforEuclid'
saxioms;
arithmeticisamodelforPeano'
saxioms.However,anysetofaxiomsalsohasother,unintendedmodels.Thesemodelsserveasillustrationsorinterpretationsoftheinitialsystemofabstractaxioms.
4.Theoreticalmodels.Theseareimaginativementalconstructsinventedtoaccountforobservedphysicalphenomena.Theyareusuallymechanismsorprocesses,postulatedbyanalogywithfamiliarmechanismsorprocesses(e.g.,thebilliardballmodelforgas,Maxwell’smechanicalmodelforelectro-magneticforces,etc.).Theirchiefuseistohelpusunderstandtheworldratherthanmerelytomakepredictions.Theyaresymbolicrepresentationsofphysicalsystems,aimingtorepresenttheunderlyingstructureoftheworld.Atheoreticalmodelisaframeworkofideasandconceptsfromwhichweinterpretourobservationsandexperimentalresults.Initshighestform,aphysicalmodelisexpressedasasetofnaturallaws(e.g.Newton'
slawsofmotion).Thephysicaltheoryshouldbebasedonaminimumnumberofphysicalassumptions,and,ideally,mustbroadenourunderstandingofthephysicalphenomenon.
Amathematicalmodelisusuallyabuilt-inelementofatheoreticalmodel.Thephysicaltheoryinterpretsthemathematicalmodel,includingitsassumptionsandconstraints.Themathematicalmodelisneededtoquantifythephysicalmodel,thusenablingthetheoreticalmodeltomakeprecisepredictionsandapplications.
3.ModelsinHistoricalCosmology
Weshallconsiderfirsttheroleofmathematicalmodelsincosmology.Aprimegoalhereistogiveasimplifiedrepresentationofthephysicaluniverseintermsofafewbasictheoreticalpresuppositions.AncientGreekcosmology—particularlythatofPlatoandAristotle—wasgenerallygeocentric.ThecelestialobjectscircledaroundafixedEarth.Uniformcircularmotionwasthoughttoreflecttheperfectionoftheheavens.Theearliesttheoreticalmodelsconsistedoftheplanetsembeddedinglassshells.ThePrimeMover,situatedbeyondtheoutershell,settheoutershellinmotion.Fromthere,motionwasmechanicallytransmittedinwardstotheothershells.
Manyancientcosmologicalmodelsreflectedalsotheologicaltruths.Inmedievalcosmologyheaven,forexample,wasplacedbeyondtheoutersphere(seeFigure1A).Dante’stheological/physicalmodel(Figure1B)includedHell,withitsvariouslevels,deepinsidetheEarth.Suchmodelsinvolvedapresentationoftheologicaltruthsanddidnotaimatanaccuratequantitativedescription.
Platoalreadynotedthatthesesimplemodelsdidnotquiterepresenttheactualmotionoftheplanets,whichdifferedsignificantlyfromuniformcircularmotion.HencePlatosethisstudentsthetaskofdevisingimprovedmathematicalmodelsthatwouldbetter'
savethephenomena.'
Theaimwastogiveanaccuratecosmologicaldescriptionbaseduponuniformcircularmotionsinthecelestialrealm.Aristotletriedtosolvetheproblembyusing55glassintermediaryspheresbutthissophisticatedmodelstillfellshortoftheobservations.
AB
Figure1.MedievalCosmologicalModels
Ptolemy'
sEpicyclesandInstrumentalism
TheproblemwaseventuallysolvedbyClaudiusPtolemy,around150A.D.Heinventedanumberofnovelgeometricaldevices.Theseincludedtheepicycle(asmallcirclesuperimposeduponalargercircle,calledthedeferent),theeccentric(adevicemakingthecenterofthecirclerotateoff-centerabouttheEarth),andtheequant(anotheroff-centerpointfromwhichspeedswerecalculated,inordertoretainuniformspeeds.TheseconceptsaredepictedinFigure2.Theresultinggeometricmodelworkedverywell.Ityieldedresultsthatcloselyapproximatedtheobservedmotions,thusenablingastronomerstopredictfutureplanetarypositions.However,inthecaseofsomeplanetsitwasfoundnecessarytoaddanotherlayerofepicycles,smallerepicyclesmovingaboutlargerones,toadequatelydescribetheobservedmotions.ThecompletePtolemaicsystemconsistedof40epicycles.
AlthoughPtolemy'
smodelgaveusefulpredictions,itwasapurelymathematicalmodelwithnophysicalunderpinnings,otherthansatisfyingtheclassicaldemandforacombinationofuniform,circularmotions.Ptolemydefendedhismathematicalmodelbyadoptingananti-realist(i.e.,instrumentalist)viewofmathematicalmodels.Hepresentedhismodelasjustausefulfictionforpracticalprediction.Hisprimecriteriawere
(1)accuracyinsavingtheappearancesand
(2)maximumsimplicity.HisapproachofaddinglayersofepicyclescorrespondstothemodernapplicationofaFourierseries.
Figure2.EpicyclicTheory.Aplanetrevolvesaboutasmallcircle,anepicycle,whichinturnrevolvesaboutalargercircle,thedeferent.ThedeferentiseccentricwhenitscenterisnotattheEarth.Theequantisanon-centralpointaboutwhichtheepicyclemovesataconstantangularrate
Thisviewofscientifictheorizingwasquitedifferentfromtherival,"
realist"
positionthathadbeendefendedbyAristotle,whobelievedthattheoriesshoulddomorethanmerelyfittheobservations.Aristotleinsistedthatmodelsshouldbeinaccordwiththetruenatureofthings.ThushisfollowersrejectedPtolemy'
ssystemsinceitcontradictedtheprinciplesofAristotle'
sphysics.
CopernicusandthequestionoftheEarth'
smotion
TheproperfunctionofcosmologicalmodelswasmuchdiscussedwiththeadventofCopernicus'
(1543)heliocentricmodel.Copernicus’modelwasapurelygeometricone,withnobasisinphysics.Itkepttopurecircles(withepicycles)andwasnosimplerthanPtolemy'
smodel.However,itdidexplainthehithertopuzzlingretrogrademotionofplanets.Thiswasnowseenasonlyanapparenteffect,duetotherelativemotionsoftheEarthandouterplanetsabouttheSun.AlthoughCopernicusbelievedhismodeltobeatruerepresentationofreality,theforwardtohisbook(writtenbytheLutherantheologianOsiander)presenteditasmerelyausefulcalculatingdevice.
Afewyearslater(in1588)TychoBrahepresentedanalternativemodelinwhichalltheplanetsrevolvedaroundtheSun,whichinturnmovedaboutafixedEarth.ThismodelwasobservationallyequivalenttothatofCopernicus.How,then,wasonetochoosebetweenthem?
Tychoheldthat,sincehismodelaccordedwiththegeocentriccosmologyoftheBible,itwastheologicallypreferable.
TheCopernicanmodelwasnotwidelyacceptedasatruedepictionofrealityuntilabout1650,whenaphysicalbasiswasfinallyfoundforit,intheformofDescartes'
theoryofvortices.Descartes’theorywassoonfalsifiedbyNewton(in1689).Newtonianmechanics,onthebasisofitsinertialframesofreference,supportedauniverseinwhichboththeEarthandtheSunwere
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