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1、MathematicalIntuitionandtheCognitiveRootsofMathematicalConceptsGiuseppeLongo•ArnaudViarougePublishedonline:20January2010SpringerScience+BusinessMediaB.V.2010AbstractThefoundationofMathematicsisbothalogicaformalissueandanepistemologicalone.Bythefirst,wemeantheexplicitationandanalysisofformalp
2、roofprinciples,which,largelyaposteriori,groundproofongeneraldeductionrulesandschemata.Bythesecond,wemeantheinvestigationoftheconstitutivegenesisofconceptsandstructures,theaimofthispaper.This‘‘genealogyofconcepts’’,isnecessarybothinordertoenrichthefoundationalanalysiswithanoftendisregardedas
3、pect(thecognitiveandhistoricalconstitutionofmathematicalstructures)andbecauseoftheprovableincompletenessofproofprinciplesalsointheanalysisofdeduction.Forthepurposesofourinvestigation,wewillhintheretoaphilosophicalframeaswellastosomerecentexperimentalstudiesonnumericalcognitionthatsupportour
4、claimonthecognitiveoriginandtheconstitutiveroleofmathematicalintuition.Keywords:NumericalcognitionMathematicalintuitionFoundationsofmathematics1FromLogictoCognitionOverthecourseofthetwentiethcentury,therelationshipsbetweenPhilosophyandMathematicshavebeendominatedbyMathematicalLogic.Amosti
5、nterestingareaofMathematicswhich,from1931onwards,yearofoneofthemajormathematicalresultsofthecentury(Go¨delianIncompleteness),enjoyedthedoublestatusofadisciplinethatis11bothtechnicallyprofoundandphilosophicallyfundamental.Fromthefoundationalpointofview,ProofTheoryconstituteditsmainaspect,als
6、oonaccountofotherremarkableresults(OrdinalAnalysis,TypeTheoryinthemannerofChurch-Go¨del-Girard,variousformsofincompleteness-independenceinSetTheoryandArithmetics),andproducedspin-offswithfar-reachingpracticalconsequences:thefunctionsforthecomputationofproofs(Herbrand,Go¨del,Church),theLogic
7、alComputingMachine(Turing),andhenceourdigitalmachines.Thequestionshavingarisenattheendofthenineteenthcentury,duetothefoundationaldebacleofEuclideancertitudes,motivatedthecentralityoftheanalysisofproofs.Inparticular,theinvestigationoftheformalconsistencyo
