外文翻译数学直觉和认知的根源.docx
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外文翻译数学直觉和认知的根源.docx
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外文翻译数学直觉和认知的根源
MathematicalIntuitionandtheCognitiveRootsofMathematical
Concepts
GiuseppeLongo•ArnaudViarouge
Publishedonline:
20January2010
SpringerScience+BusinessMediaB.V.2010
Abstract
ThefoundationofMathematicsisbothalogicaformalissueandanepistemologicalone.Bythefirst,wemeantheexplicitationandanalysisofformalproofprinciples,which,largelyaposteriori,groundproofongeneraldeductionrulesandschemata.Bythesecond,wemeantheinvestigationoftheconstitutivegenesisofconceptsandstructures,theaimofthispaper.This‘‘genealogyofconcepts’’,isnecessarybothinordertoenrichthefoundationalanalysiswithanoftendisregardedaspect(thecognitiveandhistoricalconstitutionofmathematicalstructures)andbecauseoftheprovableincompletenessofproofprinciplesalsointheanalysisofdeduction.Forthepurposesofourinvestigation,wewillhintheretoaphilosophicalframeaswellastosomerecentexperimentalstudiesonnumericalcognitionthatsupportourclaimonthecognitiveoriginandtheconstitutiveroleofmathematicalintuition.
Keywords:
Numericalcognition_
Mathematicalintuition_Foundationsofmathematics
1FromLogictoCognition
Overthecourseofthetwentiethcentury,therelationshipsbetweenPhilosophyandMathematicshavebeendominatedbyMathematicalLogic.AmostinterestingareaofMathematicswhich,from1931onwards,yearofoneofthemajormathematicalresultsofthecentury(Go¨delianIncompleteness),enjoyedthedoublestatusofadisciplinethatisbothtechnicallyprofoundandphilosophicallyfundamental.Fromthefoundationalpointofview,ProofTheoryconstituteditsmainaspect,alsoonaccountofotherremarkableresults(OrdinalAnalysis,TypeTheoryinthemannerofChurch-Go¨del-Girard,variousformsofincompleteness-independenceinSetTheoryandArithmetics),andproducedspin-offswithfar-reachingpracticalconsequences:
thefunctionsforthecomputationofproofs(Herbrand,Go¨del,Church),theLogicalComputingMachine(Turing),andhenceourdigitalmachines.
Thequestionshavingarisenattheendofthenineteenthcentury,duetothefoundationaldebacleofEuclideancertitudes,motivatedthecentralityoftheanalysisofproofs.Inparticular,theinvestigationoftheformalconsistencyofArithmetics(asFormalNumberTheory,doesityieldcontradictions?
),andofthe(non-euclidean)geometriesthatcanbeencodedbyanalytictoolsinArithmetics(allofthem—Hilbert1899—:
aretheyatleastconsistent?
).Formany,duringthetwentiethcentury,alloffoundationalanalysiscouldbereducedtothespilloversproducedbythesemajortechnicalquestions(provableconsistencyandcompleteness),broughttothelimelight,byanimmensemathematician,Hilbert.
AndhereweforgetthatinMathematics,ifitisnecessarytoproduceproofs,askeypartofthemathematician’sjob,themathematicalactivityisfirstofallgroundedonthepropositionorontheconstructionofconceptsandstructures.Infact,anyslightlyoriginalproofrequirestheinventionofnewconceptsandstructures;thepurelydeductivecomponentwillfollow.Now,asthekeenest
amongthefoundingfatherswouldsay,let’sputasidethis‘‘heuristic’’andratherfocusourattentionontheaposteriorireconstructionofthelogicalcertitudeofproof.Anabsolutelyindispensableprogram,asweweresaying,atthebeginningofthetwentiethcentury,followingthe
technicalrichnessaswellasconfusionofthenineteenth—acenturyhavingproducedmanyresults,amongwhichseveralwerefalseorunprovenorbadlystated—aprogram,however,whichexcludedfromfoundationalanalysisanyscientificexaminationoftheconstitutiveprocessofmathematicalconceptsandstructures.SuchistheaimofthenewprojectregardingthecognitivefoundationsofMathematics,whichisalsoanepistemologicalproject.Itiscertainlynotaquestionofdiscardingproof,withitslogicalandformalcomponents,butsimplyofsteeringawayfromtheformal(andcomputational)monomania,formerlyjustifiedandwhichdominatedthepreviouscentury.Itproducedthesemarvelouslogicaformalmachinessurroundingus,actingwithoutmeaning.TheanalysisoftheconstitutionofsenseandmeaninginMathematics,throughcognitionandhistory,isthepurposeoftheinvestigationsinMathematicalCognition,alsoinordertocompensatetheprovableincompletenessofformalisms.
Inordertograspthisissue,itisnecessarytobepreciseregardingtheterm‘‘formal’’,propertotheformalsystemsextensivelytakentobetheonlylocusforfoundation.Thereexistsaverywidespreadambiguity,particularlyinthefieldofphysics:
foraphysicist,to‘‘formalize’’aphysicalprocessmeanstomathematizeit.SinceHilbert’sprogram,formalinsteadhasmeant:
asystemgivenbyfinitesequencesofmeaninglesssigns,governedbyrules,themselvesbeingfinitesequencesofsigns,whichonlyoperatebypurelymechanical‘‘sequence-matching’’and‘‘sequence-replacement’’—asdolambda-calculusandTuringmachines,forinstance,twoparadigmsforanyeffectiveformalism,followingtheequivalenceresults.Tobefair,somerecent
revitalizationofHilbert’sprogramtrytoextendthisnotionof‘‘formal’’,sometimesinreferencetowritingsbyHilbertaswell.Onemaysurelyproposenewnotions,yetthedefinitionofwhataformalsystemis,isformallygivenbyincompletenesstheoremandTuring’swork.As
Go¨delsaysinhis‘‘addednote’’of1963,wehave,sinceTuring,a‘‘certain,precise,adequatenotionoftheconceptofformalsystem’’.And,notonlyTuringMachines,but,westress,alsotheincompletenesstheorem,byitsuseofthesamenotionofformalsystemmakesitperfectlystable:
definitions,inmathematics,aredefinitelystabilizedbythe(important)theoremswheretheyapply.
Ofcourse,thereiscircularityinanyformaldefinitionofaformalsystem,inthesamewaythereiscircularitywhenthenotionofwordisdefinedbymeansofotherwords:
TuringMachinesalsoneedtobedefinedasaformalsystem(yet,averybasicone).Nothingserious,weareaccustomedtothis,justaswhenwedonotstopspeaking,andwecontinuetotalkaboutsentencesusingsentences.Butthereisfarmorethanthat:
theformalistfoundationofMathematicsreferstotheabsolutecertaintyofnotionssuchas‘‘finite’’and‘‘discrete’’.Yet,asforfiniteness,followingtheOver-spillLemmainArithmeticsand,morenotably,sinceincompleteness,weknowthatwecannotformallydefinethenotionof‘‘finite’’.Infact,anaxiomofinfinityisrequiredinordertoformally‘‘isolate’’thestandardfiniteintegers.Aresultwhichdemolishesthecoreofanyfinitist
certitude,includingthatwhichisinternaltologicalfor-malism.Inshort,thenotionoffinitudeisverycomplex,requiresinfinity,ifonewantstograspitformally:
itisfarfrombeing‘‘obvious’’,intheCartesiansense.Anditisnotanabsolute:
finitenessmakeslittleabsolutesense,forinstance,incosmology—considerthequestion:
IstheRelativisticUniversefiniteorinfinite?
TheRiemannsphereisfinitebutunlimitedagainsttheGreekidentificationbetweenfiniteandlimited.Likewiseforthenotionsofcontinuity(inthemannerofCantor–Dedekind,typically)andofdiscreteness(thelattertobedefinedasanystructureofwhichthediscretetopologyis‘‘natural’’,wewouldsay
fromtheviewpointofMathematics):
DotheymakesenseinQuantumMechanics,wherethediscreteenergyspectrumhasasacounterpart,inspace,non-locality,non-separabilityproperties,theoppositeofthoseofdiscretetopologywhosepointsarewellseparated,andwhereamoreadaptedmathematicalcontinuumshould,perhaps,notbemadeofCantorianpointseither?
Asregardstheformal/mathematical,physicistsmayverywellpreservetheirambiguityoflanguage,wheretheformalisidentifiedwiththemathematical(‘‘formal’’structuresinsteadof‘‘mathematical’’structures,aswesaid),oncetheissuehasbeenclarified;because,onlythroughthisdistinctionmaytherecentresultsofthemathematicalincompletenessofformalismsbeunderstood.
Inshort,inthedifferencebetweentheformalandthemathematical,thereisnolessthansomeofthemostimportantresultsinProofTheoryofthelast30years(seeParisandHarrington(1978),seeLongo(2002;2005)forsurveysandreflections).Inthisdifferenceactuallyliesthestructuralsignificanceofintegers,theseentitiesconstitutedwithinourcognitiveandhistoricalspaces,theissueunderdiscussioninthistext.
2CognitionandInvariance
BeginningwithMathematics’CognitiveAnalysis,theperspectiveassumedhere,westresstheepistemologicalcontentofthisinvestigation.Inourview,anyepistemologyshouldalsorefertothe‘‘genealogyofconcepts’’followingRiemann.Whatcanbesaid,forinstance,abouttheconceptofinfinitywithoutmakingreferencetoitshistory?
Theiteratedgestureandthemetaphoricmappingofitsconclusion(thelimit)areinformativeideas,yettheydonotsuffice.Thereisnoconstitutivehistoryofthisconceptwithoutananalysisofthehistoricaldebatethatbrought(actual)infinitytotoday’srobustconceptualstatus,fromAristotletoSaintThomasandItalianRenaissancePaintinguptoCantor.LikewiseforrealnumbersinCantor’scontinuum:
theirobjectivityisintheirconstruction,whichistheresultofahistoricalpractice,asistheireffectiveness.
Wenowarriveatacrucialpoint;numerousauthorsrefertothe‘‘greatstabilityandreliability’’ofMathematicswhichwouldneedtobeaccountedfor.Wigner’sarticle,whicheverybodyquotes—duetoitssoveryeffectiveandmemorabletitle—andwhichveryfewread,presentsexamplesthatarenotastounding).
Dolinguists(cognitivists,forexample)considerthefollowingproblem:
What
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